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8.2: Identify and Simplify Roots


Learning Objectives

  • Square Roots
    • Use square root notation to write principal square roots
    • Simplify principal square roots using factorization
  • Cube Roots
    • Use cube root notation to write cube roots
    • Simplify cube roots using factorization
  • Simplify Square Roots
    • Simplify square roots with variables
    • Determine when a simplified root needs an absolute value
  • Rational Exponents
    • Convert between radical and exponent notation
    • Use the laws of exponents to simplify expressions with rational exponents
    • Use rational exponents to simplify radical expressions

We know how to square a number:

(left(-5 ight)^2=25)

Taking a square root is the opposite of squaring so we can make these statements:

  • 5 is the nonngeative square root of 25
  • -5 is the negative square root of 25

Find the square roots of the following numbers:

  1. 36
  2. 81
  3. -49
  4. 0
  1. We want to find a number whose square is 36. (6^2=36) therefore, the nonnegative square root of 36 is 6 and the negative square root of 36 is -6
  2. We want to find a number whose square is 81. (9^2=81) therefore, the nonnegative square root of 81 is 9 and the negative square root of 81 is -9
  3. We want to find a number whose square is -49. When you square a real number, the result is always positive. Stop and think about that for a second. A negative number times itself is positive, and a positive number times itself is positive. Therefore, -49 does not have square roots, there are no real number solutions to this question.
  4. We want to find a number whose square is 0. (0^2=0) therefore, the nonnegative square root of 0 is 0. We do not assign 0 a sign, so it has only one square root, and that is 0.

The notation that we use to express a square root for any real number, a, is as follows:

Writing a Square Root

The symbol for the square root is called a radical symbol. For a real number, a the square root of a is written as (sqrt{a})

The number that is written under the radical symbol is called the radicand.

By definition, the square root symbol, (sqrt{hphantom{5}}) always means to find the nonnegative root, called the principal root.

(sqrt{a}) is defined for (a>0)

Let’s do an example similar to the example from above, this time using square root notation. Note that using the square root notation means that you are only finding the principal root – the nonnegative root.

Example

Simplify the following square roots:

  1. (sqrt{16})
  2. (sqrt{9})
  3. (sqrt{-9})
  4. (sqrt{5^2})

[reveal-answer q=”614386″]Show Solution[/reveal-answer]
[hidden-answer a=”614386″]

  1. (sqrt{16}=4). We only write the nonnegative root because that is how the root symbol is defined.
  2. (sqrt{9}=3). We only write the nonnegative root because that is how the root symbol is defined.
  3. (sqrt{-9}). We are looking for a number whose square is -9. There are no real numbers whose square is -9, so this radical is not a real number.
  4. (5^2). We already have the number whose square is (5^2), it’s 5!

[/hidden-answer]

The last problem in the previous example shows us an important relationship between squares and square roots, and we can summarize it as follows:

The square root of a square

For a nonnegative real number, a, (sqrt{a^2}=a)

In the video that follows, we simplify more square roots using the fact that (sqrt{a^2}=a) means finding the principal square root.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=140

What if you are working with a number whose square you do not know right away? We can use factoring and the product rule for square roots to find square roots such as (sqrt{225}).

The Product Rule for Square Roots

Given that a and b are nonnegative real numbers, (sqrt{acdot{b}}=sqrt{a}cdotsqrt{b})

In the examples that follow we will bring together these ideas to simplify square roots of numbers that are not obvious at first glance:

  • square root of a square,
  • the product rule for square roots
  • factoring

Example

Simplify (sqrt{144})

[reveal-answer q=”620082″]Show Solution[/reveal-answer]
[hidden-answer a=”620082″]

Determine the prime factors of 144.

(egin{array}{c}sqrt{144}\sqrt{2cdot 72}\sqrt{2cdot 2cdot 36}\sqrt{2cdot 2cdot 2cdot 18}\sqrt{2cdot 2cdot 2cdot 2cdot 9}\sqrt{2cdot 2cdot 2cdot 2cdot 3cdot 3}end{array})

Because we are finding a square root, we regroup these factors into squares.

(sqrt{2^2cdot 2^2cdot3^2})

Now we can use the product rule for square roots and the square root of a square idea to finish finding the square root.

(egin{array}{c}sqrt{2^2cdot 2^2cdot3^2}=sqrt{2^2}cdotsqrt{2^2}cdotsqrt{3^2}=2cdot3cdot2=12end{array})

Answer

(sqrt{144}=12)

[/hidden-answer]

Example

Simplify (sqrt{225})
[reveal-answer q=”686109″]Show Solution[/reveal-answer]
[hidden-answer a=”686109″]

First, factor 225:

(egin{array}{c}sqrt{225}=sqrt{5cdot45}=sqrt{5cdot5cdot9}=sqrt{5cdot5cdot3cdot3}end{array})

Because we are finding a square root, we regroup these factors into squares. Finish simplifying with the product rule for roots, and the square of a square idea.

(egin{array}{c}sqrt{5^2cdot3^2}=sqrt{5^2}cdotsqrt{3^2}=5cdot3=15end{array})

Answer

(sqrt{225}=15)

[/hidden-answer]

Caution! The square root of a product rule applies when you have multiplication ONLY under the square root. You cannot apply the rule to sums:

(sqrt{a+b} esqrt{a}+sqrt{b})

Prove this to yourself with some real numbers: let a = 64 and b = 36, then use the order of operations to simplify each expression.

(egin{array}{c}sqrt{64+36}=sqrt{100}=10\sqrt{64}+sqrt{36}=8+6=1410 e14end{array})

So far, you have seen examples that are perfect squares. That is, each is a number whose square root is an integer. But many radical expressions are not perfect squares. Some of these radicals can still be simplified by finding perfect square factors. The example below illustrates how to factor the radicand, looking for pairs of factors that can be expressed as a square.

Example

Simplify. (sqrt{63})

[reveal-answer q=”908978″]Show Solution[/reveal-answer]
[hidden-answer a=”908978″]Factor 63

(sqrt{7cdot 3cdot3})

Regroup factors into squares

(sqrt{7cdot3^2})

Finish simplifying with the product rule for roots, and the square of a square idea.

(sqrt{7cdot3^2}=sqrt{7}cdotsqrt{3^2}=sqrt{7}cdot3)

Since 7 is prime and we can’t write it as a square, it will have to stay under the radical sign. As a matter of convention, we write the constant, 3, in front of the radical. This helps the reader know that the 3 is not under the radical anymore.

(3cdot sqrt{7})

Answer

(sqrt{63}=3sqrt{7})

[/hidden-answer]

The final answer (3sqrt{7}) may look a bit odd, but it is in simplified form. You can read this as “three radical seven” or “three times the square root of seven.”

Shortcut This Way

In the next example, we take a bit of a shortcut by making use of the common squares we know, instead of using prime factors. It helps to have the squares of the numbers between 0 and 10 fresh in your mind to make simplifying radicals faster.

  • (0^2=0)
  • (2^2=4)
  • (3^2=9)
  • (4^2=16)
  • (5^2=25)
  • (6^2=36)
  • (7^2=49)
  • (8^2=64)
  • (9^2=81)
  • (10^2=100)

Example

Simplify. (sqrt{2,000})

[reveal-answer q=”932245″]Show Solution[/reveal-answer]
[hidden-answer a=”932245″]Factor 2,000 to find perfect squares.

(egin{array}{r}sqrt{100cdot 20}=sqrt{100cdot 4cdot 5}end{array})

(100=10^2,4=2^2)

(egin{array}sqrt{100cdot 4cdot 5}= sqrt{10^2cdot 4^2cdot 5}=sqrt{10^2}cdotsqrt{4^2}cdotsqrt{5}=10cdot4cdotsqrt{5})

Multiply.

(20cdot sqrt{5})

Answer

(sqrt{2,000}=20sqrt{5})

[/hidden-answer]

In this last video, we show examples of simplifying radicals that are not perfect squares.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=140

Cube Roots

Rubik’s Cune

While square roots are probably the most common radical, you can also find the third root, the fifth root, the 10th root, or really any other nth root of a number. Just as the square root is a number that, when squared, gives the radicand, the cube root is a number that, when cubed, gives the radicand.

Find the cube roots of the following numbers:

  1. 27
  2. 8
  3. -8
  4. 0
  1. We want to find a number whose cube is 27. (9=3^2), so (3/cdot3/cdot3=3^3=27)
  2. We want to find a number whose cube is 8. (2cdot2cdot2=8) the cube root of 8 is 2.
  3. We want to find a number whose cube is -8. We know 2 is the cube root of 8, so maybe we can try -2. (-2cdot{-2}cdot{-2}=-8), so the cube root of -8 is -2. This is different from square roots because multiplying three negative numbers together results in a negative number.
  4. We want to find a number whose cube is 0. (0) by itself, you will always get (0).

The cube root of a number is written with a small number 3, called the index, just outside and above the radical symbol. It looks like (sqrt[3]). This little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.

(3sqrt{x}), three times the square root of x. They may look similar at first, but they lead you to much different expressions!

We can also use factoring to simplify cube roots such as (sqrt[3]{125}). You can read this as “the third root of 125” or “the cube root of 125.” To simplify this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. Let’s factor 125 and find that number.

Example

Simplify. (sqrt[3]{125})

[reveal-answer q=”517592″]Show Solution[/reveal-answer]
[hidden-answer a=”517592″]125 ends in 5, so you know that 5 is a factor. Expand 125 into (5cdot25).

(sqrt[3]{5cdot 25})

Factor 25 into 5 and 5.

(sqrt[3]{5cdot 5cdot 5})

The factors are (5^{3}).

(sqrt[3]

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)

Answer

(sqrt[3]{125}=5)

[/hidden-answer]

The prime factors of 125 are (5^{3}). The cube root of a cubed number is the number itself, so (sqrt[3]

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=5). You have found the cube root, the three identical factors that when multiplied together give 125. 125 is known as a perfect cube because its cube root is an integer.

Here’s an example of how to simplify a radical that is not a perfect cube.

Example

Simplify. (sqrt[3]{32

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})

[reveal-answer q=”617053″]Show Solution[/reveal-answer]
[hidden-answer a=”617053″]Factor 32 into prime factors.

(sqrt[3]{2cdot 2cdot 2cdot 2cdot 2cdot

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})

Since you are looking for the cube root, you need to find factors that appear 3 times under the radical. Rewrite (

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).

(sqrt[3]

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)

Rewrite (

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cdot

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).

(sqrt[3]

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)

Rewrite the expression as a product of multiple radicals.

(sqrt[3]

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cdot sqrt[3]{2cdot 2}cdot sqrt[3]

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cdot sqrt[3]

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)

Simplify and multiply.

(2cdot sqrt[3]{4}cdot mcdot sqrt[3]

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)

Answer

(sqrt[3]{32

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}=2msqrt[3]{4

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})

[/hidden-answer]

In the example below, we use the following idea:

(sqrt[3]

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=-1)

to simplify the radical. You do not have to do this, but it may help you recognize cubes more easily when they are nonnegative.

Example

Simplify. (sqrt[3]{-27

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})

[reveal-answer q=”670300″]Show Solution[/reveal-answer]
[hidden-answer a=”670300″]Factor the expression into cubes.

Separate the cubed factors into individual radicals.

(egin{array}{r}sqrt[3]{-1cdot 27cdot

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cdot

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}sqrt[3]

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sqrt[3]

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cdot sqrt[3]

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cdot sqrt[3]

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cdot sqrt[3]{x}cdot sqrt[3]

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end{array})

Simplify the cube roots.

(-1cdot 3cdot xcdot ycdot sqrt[3]{x})

Answer

(sqrt[3]{-27

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}=-3xysqrt[3]{x})

[/hidden-answer]

In the video that follows, we show more examples if simplifying cube roots.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=140

You could check your answer by performing the inverse operation. If you are right, when you cube (-27

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).

(egin{array}{l}left( -3xysqrt[3]{x} ight)left( -3xysqrt[3]{x} ight)left( -3xysqrt[3]{x} ight)-3cdot -3cdot -3cdot xcdot xcdot xcdot ycdot ycdot ycdot sqrt[3]{x}cdot sqrt[3]{x}cdot sqrt[3]{x}-27cdot

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cdot

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cdot sqrt[3]

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\-27

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cdot x-27

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end{array})

You can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can simplify the radicals (sqrt[7]{-2187}), but you cannot simplify the radicals (sqrt[6]{-2,500}).

Let’s look at another example.

Example

Simplify. (sqrt[3]{-24

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})

[reveal-answer q=”473861″]Show Solution[/reveal-answer]
[hidden-answer a=”473861″]Factor (−1) and 8 are the perfect cubes.

(sqrt[3]{-1cdot 8cdot 3cdot

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})

Factor variables. You are looking for cube exponents, so you factor (a^{3}) and (a^{2}).

(sqrt[3]

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)

Separate the factors into individual radicals.

(sqrt[3]

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cdot sqrt[3]

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cdot sqrt[3]

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cdot sqrt[3]{3cdot

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})

Simplify, using the property (sqrt[3]

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=x).

(-1cdot 2cdot acdot sqrt[3]{3cdot

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})

This is the simplest form of this expression; all cubes have been pulled out of the radical expression.

(-2asqrt[3]{3

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})

Answer

(sqrt[3]{-24

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}=-2asqrt[3]{3

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})

[/hidden-answer]

The steps to consider when simplifying a radical are outlined below.

Simplifying a radical

When working with exponents and radicals:

  • If n is odd, (sqrt[n]

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    =x).
  • If n is even, (sqrt[n]

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    =left| x ight|). (The absolute value accounts for the fact that if x is negative and raised to an even power, that number will be positive, as will the nth principal root of that number.)

Example

Simplify. (sqrt{100

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})

[reveal-answer q=”982628″]Show Solution[/reveal-answer]
[hidden-answer a=”982628″]Separate factors; look for squared numbers and variables. Factor 100 into (10cdot10).

(sqrt{10cdot 10cdot

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cdot

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})

Factor (left(y^{2} ight)^{2}).

(sqrt{10cdot 10cdot

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cdot

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})

Separate the squared factors into individual radicals.

(sqrt

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cdot sqrt

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cdot sqrt

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)

Take the square root of each radical . Since you do not know whether x is positive or negative, use (left|x ight|) to account for both possibilities, thereby guaranteeing that your answer will be positive.

(10cdotleft|x ight|cdot{y}^{2})

Simplify and multiply.

(10left|x ight|y^{2})

Answer

(sqrt{100

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}=10left| x ight|

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)

[/hidden-answer]

You can check your answer by squaring it to be sure it equals (100

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).

In the last video, we share examples of finding cube roots with negative radicands.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=140

Simplify Square Roots with Variables

Radical expressions are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as(sqrt[3]{250

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y}). Using factoring, you can simplify these radical expressions, too.

Radical

Simplifying Square Roots

Radical expressions will sometimes include variables as well as numbers. Consider the expression (sqrt{9

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}). Simplifying a radical expression with variables is not as straightforward as the examples we have already shown with integers.

Consider the expression (sqrt

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). This looks like it should be equal to x, right? Let’s test some values for x and see what happens.

In the chart below, look along each row and determine whether the value of x is the same as the value of (sqrt

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). Where are they equal? Where are they not equal?

After doing that for each row, look again and determine whether the value of (left|x ight|).

(x^{2})(left|x ight|)
(−2)422
0000
63666
101001010

Notice—in cases where x is a negative number, (sqrt{x^{2}}=left|x ight|). You need to consider this fact when simplifying radicals that contain variables, because by definition (sqrt{x^{2}}) is always nonnegative.

Taking the Square Root of a Radical Expression

When finding the square root of an expression that contains variables raised to a power, consider that (sqrt{x^{2}}=left|x ight|).

Examples: (sqrt{16

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}=4left|xy ight|)

Let’s try it.
The goal is to find factors under the radical that are perfect squares so that you can take their square root.

Example

Simplify. (sqrt{9

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})

[reveal-answer q=”41297″]Show Solution[/reveal-answer]
[hidden-answer a=”41297″]Factor to find identical pairs.

(sqrt{3cdot 3cdot

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cdot

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})

Rewrite the pairs as perfect squares, note how we use the power rule for exponents to simplify ({x^3}^2)

(sqrt

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)

Separate into individual radicals.

(sqrt

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cdot sqrt

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)

Simplify, using the rule that (sqrt

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=left|x ight|).

(3left|

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ight|)

Answer

(sqrt{9

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}=3left|

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ight|)

[/hidden-answer]

Variable factors with even exponents can be written as squares. In the example above, (

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=

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cdot

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={left|x^3 ight|}^{2}) and

(

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=

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cdot

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={left(|y^2 ight|)}^{2}).

Let’s try to simplify another radical expression.

Example

Simplify. (sqrt{49

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})

[reveal-answer q=”283065″]Show Solution[/reveal-answer]
[hidden-answer a=”283065″]Look for squared numbers and variables. Factor 49 into (x^{10}) into (y^{8}) into (y^{4}cdot{y}^{4}).

(sqrt{7cdot 7cdot

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cdot

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cdot

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cdot

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})

Rewrite the pairs as squares.

(sqrt

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)

Separate the squared factors into individual radicals.

(sqrt{7^2}cdotsqrt{({x^5})^2}cdotsqrt{({y^4})^2})

Take the square root of each radical using the rule that (sqrt

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=left|x ight|).

(7cdotleft|

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ight|cdot

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)

Multiply.

(7left|

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ight|

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)

Answer

(sqrt{49

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}=7left|

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ight|

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)

[/hidden-answer]

You find that the square root of (7left|

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ight|

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). In order to check this calculation, you could square (49

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). And, in fact, you would get this expression if you evaluated ({left({7left|

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ight|

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} ight)^{2}}).

In the video that follows we show several examples of simplifying radicals with variables.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=140

Example

Simplify. (sqrt

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)

[reveal-answer q=”141094″]Show Solution[/reveal-answer]
[hidden-answer a=”141094″]Factor to find variables with even exponents.

(sqrt

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)

Rewrite (left(b^{2} ight)^{2}).

(sqrt

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)

Separate the squared factors into individual radicals.

(sqrt

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cdotsqrt

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cdotsqrt

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cdot sqrt{acdot b})

Take the square root of each radical. Remember that (sqrt

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=left| a ight|).

(left| a ight|cdot

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cdotleft|{c} ight|cdotsqrt{acdot b})

Simplify and multiply. The entire quantity (b^2) will be positive anyway.

(left| a

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c ight|sqrt{ab})

Answer

(sqrt

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=left| a

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c ight|sqrt{ab})

[/hidden-answer]

In the next section, we will explore cube roots, and use the methods we have shown here to simplify them. Cube roots are unique from square roots in that it is possible to have a negative number under the root, such as (sqrt[3]{-125}).

Rational Exponents

Roots can also be expressed as fractional exponents. The square root of a number can be written with a radical symbol or by raising the number to the (frac{1}{2}) power. This is illustrated in the table below.

Exponent FormRoot FormRoot of a SquareSimplified
(sqrt{25})(

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)
(sqrt

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)
4
(sqrt{100})(sqrt

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)
10

Use the example below to familiarize yourself with the different ways to write square roots.

Example

Fill in the missing cells in the table.

Exponent FormRoot FormRoot of a SquareSimplified
(sqrt{81})
(sqrt

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)

[reveal-answer q=”990781″]Show Solution[/reveal-answer]
[hidden-answer a=”990781″]

Exponent FormRoot FormRoot of a SquareSimplified
(sqrt{36})(

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)
(sqrt

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)
9
(sqrt{144})(sqrt

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)
12

[/hidden-answer]

In the following video, we show another example of filling in a table to connect the different notation used for roots.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=140

We can extend the concept of writing (sqrt{x}=x^{frac{1}{2}}) to cube roots. Remember, cubing a number raises it to the power of three. Notice that in these examples, the denominator of the rational exponent is the number 3.

Radical Form

Exponent Form

Integer

(

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)
2
(

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)
5
(

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)
10

These examples help us model a relationship between radicals and rational exponents: namely, that the nth root of a number can be written as either (

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).

Radical Form

Exponent Form

(

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)
(

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)
(

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)
(

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)

Convert Between Radical and Exponent Notation

When faced with an expression containing a rational exponent, you can rewrite it using a radical. In the table above, notice how the denominator of the rational exponent determines the index of the root. So, an exponent of (frac{1}{5}) translates to the fifth root or (frac{1}{8}) translates to the eighth root or (sqrt[8]

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) .

Example

Write (sqrt[4]) can be rewritten as the exponent (frac{1}{4}). Remove the radical and place the exponent next to the base.

(

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)

Answer

(sqrt[3]{81}=

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)

[/hidden-answer]

Example

Express (

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) in radical form.

[reveal-answer q=”581351″]Show Solution[/reveal-answer]
[hidden-answer a=”581351″]Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.

(sqrt[3]{2x})

The parentheses in (

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) indicate that the exponent refers to everything within the parentheses.

Answer

(

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=sqrt[3]{2x})

[/hidden-answer]

Remember that exponents only refer to the quantity immediately to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one important difference—there are no parentheses! Look what happens.

Example

Express (2

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) in radical form.

[reveal-answer q=”236347″]Show Solution[/reveal-answer]
[hidden-answer a=”236347″]Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.

(2sqrt{x})

The exponent refers only to the part of the expression immediately to the left of the exponent, in this case x, but not the 2.

Answer

(2

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=2sqrt{x})

[/hidden-answer]

The next example is intended to help you practice placing a rational exponent on the appropriate terms in an expression that is written in radical form

Example

Express (4sqrt[3]{xy}) with rational exponents.

[reveal-answer q=”527560″]Show Solution[/reveal-answer]
[hidden-answer a=”527560″]Rewrite the radical using a rational exponent. The root determines the fraction. In this case, the index of the radical is 3, so the rational exponent will be (frac{1}{3}).

(4

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)

Since 4 is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.

Answer

(4sqrt[3]{xy}=4

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)

[/hidden-answer]

In the next video, we show examples of converting between radical and exponent form.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=140

When converting from radical to rational exponent notation, the degree of the root becomes the denominator of the exponent. If you start with a square root, you will have an exponent of (frac{1}{3}) you will use a cube root. The following statement summarizes this idea.

Writing Fractional Exponents

Any radical in the form (a^{frac{1}{n}}).

Simplifying Radical Expressions Using Rational Exponents and the Laws of Exponents

Let’s explore some radical expressions now and see how to simplify them. Let’s start by simplifying this expression, (sqrt[3]

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).

One method of simplifying this expression is to factor and pull out groups of (a^{3}), as shown below in this example.

Example

Simplify. (sqrt[3]

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)

[reveal-answer q=”235013″]Show Solution[/reveal-answer]
[hidden-answer a=”235013″]Rewrite by factoring out cubes.

(sqrt[3]

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)

Write each factor under its own radical and simplify.

(egin{array}{r}sqrt[3]

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cdot sqrt[3]

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acdot{a}end{array})

Answer

(sqrt[3]

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=

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)

[/hidden-answer]

You can also simplify this expression by thinking about the radical as an expression with a rational exponent, and using the principle that any radical in the form (

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).

Example

Simplify. (sqrt[3]

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)

[reveal-answer q=”898415″]Show Solution[/reveal-answer]
[hidden-answer a=”898415″]Rewrite the radical using a rational exponent.

(

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)

Simplify the exponent.

(

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)

Answer

(sqrt[3]

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=

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)

[/hidden-answer]

Note that rational exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.

Both simplification methods gave the same result, (a^{2}). Depending on the context of the problem, it may be easier to use one method or the other, but for now, you’ll note that you were able to simplify this expression more quickly using rational exponents than when using the “pull-out” method.

Let’s try another example.

Example

Simplify. (sqrt[4]{81

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ParseError: EOF expected (click for details)

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})

[reveal-answer q=”324337″]Show Solution[/reveal-answer]
[hidden-answer a=”324337″]Rewrite the radical using rational exponents.

(

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)

Use the rules of exponents to simplify the expression.

(egin{array}{r}

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cdot

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cdot

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\

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ParseError: EOF expected (click for details)

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ParseError: EOF expected (click for details)

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\

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\3

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ParseError: EOF expected (click for details)

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end{array})

Change the expression with the rational exponent back to radical form.

(3

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sqrt[4]

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)

Answer

(sqrt[4]{81

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ParseError: EOF expected (click for details)

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}=3

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sqrt[4]

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)

[/hidden-answer]

Again, the alternative method is to work on simplifying under the radical by using factoring. For the example you just solved, it looks like this.

Example

Simplify. (sqrt[4]{81

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ParseError: EOF expected (click for details)

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})

[reveal-answer q=”295348″]Show Solution[/reveal-answer]
[hidden-answer a=”295348″]Rewrite the expression.

(sqrt[4]{81}cdot sqrt[4]

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cdot sqrt[4]

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Factor each radicand.

(sqrt[4]{3cdot 3cdot 3cdot 3}cdot sqrt[4]

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Simplify.

(egin{array}{r}sqrt[4]

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3cdot

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end{array})

Answer

(sqrt[4]{81x^{8}y^{3}}=3x^{2}sqrt[4]{y^{3}})

[/hidden-answer]

The following video shows more examples of how to simplify a radical expression using rational exponents.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=140

Summary

The square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of 0 is 0. You can only take the square root of values that are nonnegative. The square root of a perfect square will be an integer. Other square roots can be simplified by identifying factors that are perfect squares and taking their square root.

A radical expression is a mathematical way of representing the nth root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property (sqrt[n]

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=left| x ight|) if n is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.

A radical can be expressed as an expression with a fractional exponent by following the convention (sqrt[n]

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). Rewriting radicals using fractional exponents can be useful in simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.


Simplifying Square Roots

To simplify a square root: make the number inside the square root as small as possible (but still a whole number):

Example: &radic12 is simpler as 2&radic3

Get your calculator and check if you want: they are both the same value!

Here is the rule: when a and b are not negative

And here is how to use it:

Example: simplify &radic12

And the square root of 4 is 2:

Example: simplify &radic8

(Because the square root of 4 is 2)

Example: simplify &radic18

It often helps to factor the numbers (into prime numbers is best):

Example: simplify &radic6 × &radic15

First we can combine the two numbers:

Then we see two 3s, and decide to "pull them out":


Square Root of 8

The square root of 8 is 2[sqrt<2>]. The square of a number, x, is obtained when it is multiplied by itself. The square root of the resulting number, x[^<2>], is expressed as [sqrt>], that is, x. Even though 8 is not a prime number, yet, when we take its square root, we get 2, as its only prime factor. Now, when we multiply an irrational number,[sqrt<2>], by a rational number, 2, the result so obtained, 2[sqrt<2>], is an irrational number. Hence, the square root of 8, i.e.,[sqrt<8>], is an irrational number 2[sqrt<2>].

The square root of 8, denoted by [sqrt<8>], is the number whose square gives you the number 8. In a simpler for, [sqrt<8>] is written as [2sqrt<2>]. This value is called surd since you cannot simplify this further. However, 8 is a perfect cube of the number 2 since 2 x 2 x 2 = 8. So, you can easily find the cube root of 8 which is denoted as [sqrt[3]<8>]= 2. However, there are a few numbers like 4, 9, 16, etc. that are perfect squares and whose squares are easy to find. In this article, we will learn about the square root of 8 and how to find the root 8 value.

[sqrt<4>] = 2, as you know 2 x 2 = 4

[sqrt<9>] = 3, as you know 3 x 3 = 9

[sqrt<16>] = 4, as you know 4 x 4 = 16

Any square root is represented by the square root symbol √. This symbol is known as the radical symbol or simply radix. The number underneath this radical symbol is known as radicand. Hence, the number whose square root is to be determined is called the radicand.


8.2: Identify and Simplify Roots

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There are a LOT of tools and techniques for this process. Here are the quality objectives related to this process

  • Implement specific design guidelines (see paragraph 8.2.2.6 on “Design for X”)
  • Implement quality assurance tools and techniques (see paragraph 8.2.2.5 on quality audits, paragraph 8.2.2.2 on Data Analysis techniques such as failure analysis)
  • Improve the efficiency and effectiveness of processes and activities to achieve better results (see paragraph 8.2.2.8 on Quality Improvement Methods)
  • Confirm that the quality processes are being implemented and that their use achieves the quality objectives (see paragraphs 8.2.2.1, 8.2.2.2, and 8.2.2.4 on data gathering, analysis, and representation techniques)

There are also some “generic” tools and techniques such as Decision Making (paragraph 8.2.2.3) and Problem Solving (paragraph 8.2.2.7) that are more general and are used with other processes in other knowledge areas.

8.2.2 Manage Quality: Tools and Techniques

  • Quality checklists–these verify that a set of required steps have been performed or to see if a list of requirements has been satisfied. These checklists are usually based on the acceptance criteria included in the scope baseline.
  • Alternatives analysis–used to identify various options or approaches with respect to quality and then to select which are most appropriate for use on this project
  • Document analysis–this analyzes the following documents to point out processes that may be out of control
    • Quality reports
    • Test reports
    • Performance reports
    • Variance analysis
    • the inputs to a process to see if there are constraints they impose on the process
    • non-value-added activities that occur during that process
    • the outputs to a process to see if they can be used to identify problems

    If there are alternatives being discussed (see “alternatives analysis” in paragraph 8.2.2.2 on Data Analysis techniques), then there needs to be a decision-making process put into place to decide among those alternatives. Having criteria in place is important so that the various alternatives can be evaluating to narrow down the alternatives that are the most viable for the project. if necessary, the final decision can be made by votes taken at a meeting of project team members and experts.

    Here are some techniques used to represent data–they are used in conjunction with the data analysis techniques listed in paragraph 8.2.2.2.

    • Affinity diagrams–these organize potential causes of defects into groups showing areas that should be focused on the most. For more information on this techique, see the post I did for the 5th Edition PMBOK just on this technique alone.
    • Cause-and-effect diagrams–these are sometimes known as fishbone diagrams, why-why diagrams, or Ishikawa diagrams, and are used to identify the root cause of the the problem being stated. This kind of diagram breaks down the potential causes of the problem statement into discrete branches. See Figure 8-9 on p. 294 of the PMBOK Guide for an example.
    • Flowcharts–used to show a series of steps that lead to a defect and are used in conjunction with process analysis (see paragraph 8.2.2.2 on Data Analysis techniques) to identify the processes that are causing the defect so that those processes can be improved using quality improvement methods (see paragraph 8.2.2.8)..
    • Histograms–these are graphical representations of historical data. Used in managing quality, they can show the number of defects per deliverable, for example. Then the deliverables that have the most defects can be identified and worked on as a priority.
    • Matrix diagrams–these show the strength of relationships among factors, causes, and objectives that exist between the rows and columns that form the matrix. For an example of a quality tool called House of Quality that uses matrix diagrams, see my post below:
    • Scatter diagrams–this shows the relationship between two variables. If there is no relationship between the two variables, there will be no discernible pattern between the data points in the diagram. If there is a positive correlation, there will be an upward-sloping pattern to the data points if there is a negative correlation, there will be a downward-sloping pattern. Do not confuse a negative correlation with no correlation at all! If X increases while Y increases, this is a positive correlation because the tendency is the same with both variables. However, if X decreases while Y increases, this is a negative correlation because the tendency is the opposite between the two variables. If there is NO correlation, then as X increases, Y may be all over the place rather than having any particular pattern.

    An audit is a structured, independent process used to determine if project activities comply with organizational and project policies, processes, and procedures. It is structured because it is done based on some sort of organized format such as a checklist (see paragraph on 8.2.2.1 Checklists). It is independent because it is not being done by the same people who are responsible for doing the project activities. This is why a quality audit is usually conducted by a team external to the project. Here are the types of things that a quality audit will look for:

    • Identification of good and best practices being implemented, and sharing of good practices being done in similar projects in the organization
    • Identification of nonconformity, gaps and shortcomings between activities as they are being done on the project and how they are supposed to be done based on project procedures set forth in the Project Management Plan.
    • Offering assistance to improve the implementation of processes (see paragraph 8.2.2.8 Quality Improvement Methods)

    Besides looking at project activities, quality audits can also confirm that approved change requests have been implemented correctly.

    8.2.2.6 Design For X (sometimes abbreviated as DfX)

    This is a set of technical guidelines that may be applied during the design of a product for the optimization of a specific aspect of the design. hat is the “X” in DfX, and it can refer to reliability, safety, usability, or any other feature of the design that you want to focus on. This is a trending topic in quality in PMI, which is designing in quality so that defects don’t get produced in the first place, rather than trying to reduce defects through inspection later on.

    Quality assurance and quality improvement requires problem solving, where you have to find the cause of the defect (the problem), generate possible solutions, choose one of those solutions, and then implement it. Finally, you have to verify that the solution is effective.


    2.2. Formal criteria to identify parts of speech

    In order to justify a classification of words into grammatical parts of speech, then, we need to develop grammatical criteria, in the sense of criteria for classification that are not tied to vague, anecdotal, or limited semantic notions.

    A well-established test of a word’s category comes from its distribution, that is, the context or set of contexts where a word can (or cannot) occur. So fundamental was this technique to the methods developed by American linguists during the first half of the 20th century that it gave name to their whole school of linguistics: Distributionalism. Distributional criteria are not used alone but in conjunction with morphological criteria, for instance, to justify the existence of a category of words or a part of speech in a language.

    Take for instance the sentence in (1a). If we remove from it the boldfaced word, we get a frame (1b), or context, into which we can insert any words we can come up with. The words in (2a), for instance, can be inserted there, but the ones in (2b) cannot.

    (1) a. I bowed my head
    b. I ______ my head
    (2) a. washed, covered, shaved, raised, injured
    b. now, dark, king, city

    The words in (2a), then, form a distributional equivalence class, which also includes bowed. The words in (2b) are not in this class. Notice too that the words in (2a) are complex, formed with a root and the affix -ed (you can apply the criteria we developed in the previous chapter to justify this). The roots of these words, too, can occur in the same slot in (1b). But what happens if we try to add an -ed affix to the words in (2b)? See for yourself!

    (3) *nowed, *darked, *kinged, *citied

    Thus, we observe a formal correlation between two seemingly independent factors: a) the ability to occur in environment (1b), and b) the ability to have an -ed affix. In order to make sense of this correlation we must postulate that all words that have this constellation of properties form a natural class, i.e. a category or part of speech. We call this category ‘verb’. We can then simply state that a verb is a word that can occur in environment (1b), and that it can have a particular affix (-ed), without having to repeat ourselves over and over again for each individual word. That is, classifying words into parts of speech helps us state regularities about the grammar, and therein lays the usefulness (and justification) for having parts of speech in our grammar. Their meaning, or notional characterization, is secondary.

    But as we learned before, it is not enough to show what a linguistic unit or set of units can do, we also need to justify our analysis by showing what it cannot do. So, Let’s look for an context or environment where the words in (2a) cannot occur. Consider the frame in (4):

    None of the words in (2a) can occur in the empty slot in (4). We can now express this fact as a regularity of the grammar: no verbs can occur in the empty slots in (4). But what about the words in (2b)? Some of them can, some of them cannot. (5) includes the words that can be inserted into (4), with the addition of a few more.

    (5) king, city, wrist, shoe, toy

    These words, then, form a separate distributional class, or part of speech. These are nouns. Nouns also have their morphological correlative: we can add an -(e)s affix to these words, getting us kings, shoes, etc. And again these words can occur in the empty slot in (4). Like our definition of ‘verb’, our definition of ‘noun’ is based only on formal (i.e. distributional, morphological) criteria. we can define ‘noun’ as a part of speech based on a constellation of distributional and morphological properties of a set of words. All of these grammatical facts would be very difficult to account for without recourse to part of speech classifications.

    Besides nouns and verbs, other major categories include adjectives and adverbs. We can identify the distributional class of adjectives with the frame in (6). The words in (7a) all fit into that frame, and they can also combine with the suffix -er, as in (7b):

    (6) my shoes are very ______.
    (7) a. dark, heavy, big
    b. darker, heavier, bigger

    finally, take an adverb like now. We can almost classify it in negative terms. Now cannot occur in any of the three environments that showed to be selective for verbs, nouns, and adjectives, respectively. Moreover, now cannot have any of the suffixes that the other word categories can have. We cannot say *nowed, *nows, or *nower!

    The internal structure of a word can also be a reliable indication of the word’s part of speech. Consider the following word list:

    writer strongly confinement simplify
    clarify blender fearless final
    colonize doubtful statement fearful
    simply painless marginal vaporize

    Words that end in -er, and -ment have the same distribution as nouns, hence they are nouns. Those ending in -ize and -ify are verbs, those ending in -ful, -less, or -al are adjectives, and those ending in -ly are adverbs.


    8.2: Identify and Simplify Roots

    Home > CCA2 > Chapter 8 > Lesson 8.2.3 > Problem 8-104

    In parts (a) through (d) below, for each polynomial function , the graph of is shown. Based on this information, state the number of linear and quadratic factors the factored form of its equation should have and how many real and complex (non-real) solutions might have. (Assume a polynomial function of the lowest possible degree for each one.)

    Example: at right will have three linear factors, therefore three real roots and no complex roots. 

    There will be three linear factors (one repeated), therefore two real (one single, one double) and zero complex (non-real) roots.

    There will be one linear factor and one quadratic factor, therefore one real and two complex (non-real) roots.

    There will be four linear factors, therefore four real and zero complex (non-real) roots.

    There will be two linear and one quadratic factor, therefore two real and two complex (non-real) roots.


    Simplify Fractions

    Use this page to reduce a fraction to it's lowest terms. Enter the numerator and denominator as integer numbers. Then press the "Simplify Fraction" button, to calculate and display the simplified fraction. You may enter positive or negative numbers for both numerator and denominator as long as their value is between -2147483648 and 2147483647. This calculator may generate improper fractions, that is, fractions that have the numerator larger than their denominator.

    How are fractions reduced to their lowest terms

    Fractions can be simplified by dividing the numerator and denominator by their greatest common factor. This is how this calculator does it.

    When reducing fractions by hand it may be easier to repeatedly divide numerator and denominator by factors that are common to both of them. The process is complete when the numerator and the denominator have no more factors in common.

    Reduced fractions are fractions that have been reduced to their simplest form. When the GCF of the numerator and denominator is equal to 1, the fraction cannot be reduced any further.


    What is Fibrous root?

    A fibrous root is a root that consists of groups of roots of similar size and length. They do not penetrate as deeply into the soil as does a taproot.

    The fibrous root is a feature of plants that are monocotyledons. Unlike the tap root, the primary root produced during development does not remain, and instead roots, known as adventitious roots, are produced from the stem of the plant.

    All these roots forming the fibrous root collectively are of equal size and length.

    A fibrous root system does not penetrate deeply into the soil but rather creates a thick network of roots that are good at holding the soil together.

    Many types of grasses have fibrous roots, including plants related to grass such as corn.

    Advantages of fibrous root systems include that they allow the plant to absorb water and minerals over a large surface area closer to the surface of the soil.

    They are also useful in helping prevent or reduce soil erosion since these root systems help hold the soil particles together.


    Using Control System Designer for Root Locus Design

    Another way to complete what was done above is to use the interactive Control System Designer tool within MATLAB. Using the same model as above, we first define the plant, .

    The controlSystemDesigner function can be used for analysis and design. In this case, we will focus on using the root locus as the design method to improve the step response of the closed-loop system. To begin, type the following into the MATLAB command window:

    The following window should appear. You can also launch the GUI by going to the APPS tab and clicking on the app icon under Control System Design and Analysis. Here you can see the root locus plot, along with open-loop Bode plot, and the closed-loop step response plot for the given plant in unity feedback with a default controller of .

    The next step is to add the design requirements to the Root Locus plot. This is done directly on the plot by right-clicking and selecting Design Requirements, New. Design requirements can be set for the Settling Time, the Percent Overshoot, the Damping Ratio, the Natural Frequency, or a Region Constraint.

    Here, we will set the design requirements for the damping ratio and the natural frequency as was done previously with the sgrid command. Recall that the boundary of our requirements call for = 0.7 and = 1.8. Set these within the design requirements. On the plot, any area which is still white is an acceptable region for the closed-loop poles.

    Zoom into the Root Locus by right-clicking on an axis and selecting Properties followed by the label Limits. Change the real-axis limits to -25 to 5 and the imaginary axis limits to -2.5 to 2.5.

    Also, we can see the current values of some key parameters in the response. On the Step response plot, right-click on the plot and go to Characteristics and select Peak Response. Repeat for the characteristic Rise Time. There should now be two large dots on the screen indicating the values of these parameters. Click each of these dots to bring up a box with information.

    Both plots should appear as shown here, once the Bode plot is closed:

    As the characteristics show on the Step response, the overshoot is acceptable, but the rise time is much too large. The pink boxes on the root locus show the corresponding closed-loop pole locations for the currently chosen control gain .

    To fix this, we need to choose a new value for the gain . Similar to how we employed the rlocfind command, the gain of the controller can be changed directly on the root locus plot. Click and drag the pink box closest to the imaginary axis (at the origin) to the acceptable region of our root locus plot as shown below.

    In the Preview box of the window, it can be seen that the loop gain has been changed to 360. Inspecting the closed-loop step response plot, both of our requirements are now met.


    Watch the video: Lesson - Simplifying Expressions with Radical Exponents (October 2021).