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4.3: Simplify Rational Exponents


Learning Objectives

By the end of this section, you will be able to:

  • Simplify expressions with (a^{frac{1}{n}})
  • Simplify expressions with (a^{frac{m}{n}})
  • Use the properties of exponents to simplify expressions with rational exponents

Before you get started, take this readiness quiz.

  1. Add: (frac{7}{15}+frac{5}{12}).
    If you missed this problem, review Example 1.28.
  2. Simplify: ((4x^{2}y^{5})^{3}).
    If you missed this problem, review Example 5.18.
  3. Simplify: (5^{−3}).
    If you missed this problem, review Example 5.14.

Simplify Expressions with (a^{frac{1}{n}})

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.

The Power Property for Exponents says that (left(a^{m} ight)^{n}=a^{m cdot n}) when (m) and (n) are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number (p) such that (left(8^{p} ight)^{3}=8). We will use the Power Property of Exponents to find the value of (p).

(left(8^{p} ight)^{3}=8)

Multiple the exponents on the left.

(8^{3p}=8)

Write the exponent (1) on the right.

(8^{3p}=8^{1})

Since the bases are the same, the exponents must be equal.

(3p=1)

Solve for (p).

(p=frac{1}{3})

So (left(8^{frac{1}{3}} ight)^{3}=8). But we know also ((sqrt[3]{8})^{3}=8). Then it must be that (8^{frac{1}{3}}=sqrt[3]{8}).

This same logic can be used for any positive integer exponent (n) to show that (a^{frac{1}{n}}=sqrt[n]{a}).

Definition (PageIndex{1}): Rational Exponent (a^{frac{1}{n}})

If (sqrt[n]{a}) is a real number and (n geq 2), then

(a^{frac{1}{n}}=sqrt[n]{a})

The denominator of the rational exponent is the index of the radical.

There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you'll practice converting expressions between these two notations.

Example (PageIndex{1})

Write as a radical expression:

  1. (x^{frac{1}{2}})
  2. (y^{frac{1}{3}})
  3. (z^{frac{1}{4}})

Solution:

We want to write each expression in the form (sqrt[n]{a}).

a.

(x^{frac{1}{2}})

The denominator of the rational exponent is (2), so the index of the radical is (2). We do not show the index when it is (2).

(sqrt{x})

b.

(y^{frac{1}{3}})

The denominator of the exponent is (3), so the index is (3).

(sqrt[3]{y})

c.

(z^{frac{1}{4}})

The denominator of the exponent is (4), so the index is (4).

(sqrt[4]{z})

Exercise (PageIndex{1})

Write as a radical expression:

  1. (t^{frac{1}{2}})
  2. (m^{frac{1}{3}})
  3. (r^{frac{1}{4}})
Answer
  1. (sqrt{t})
  2. (sqrt[3]{m})
  3. (sqrt[4]{r})

Exercise (PageIndex{2})

Write as a radical expression:

  1. (b^{frac{1}{6}})
  2. (z^{frac{1}{5}})
  3. (p^{frac{1}{4}})
Answer
  1. (sqrt[6]{b})
  2. (sqrt[5]{z})
  3. (sqrt[4]{p})

In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.

Example (PageIndex{2})

Write with a rational exponent:

  1. (sqrt{5y})
  2. (sqrt[3]{4 x})
  3. (3 sqrt[4]{5 z})

Solution:

We want to write each radical in the form (a^{frac{1}{n}})

a.

(sqrt{5y})

No index is shown, so it is (2).

The denominator of the exponent will be (2).

Put parentheses around the entire expression (5y).

((5 y)^{frac{1}{2}})

b.

(sqrt[3]{4 x})

The index is (3), so the denominator of the exponent is (3). Include parentheses ((4x)).

((4 x)^{frac{1}{3}})

c.

(3 sqrt[4]{5 z})

The index is (4), so the denominator of the exponent is (4). Put parentheses only around the (5z) since 3 is not under the radical sign.

(3(5 z)^{frac{1}{4}})

Exercise (PageIndex{3})

Write with a rational exponent:

  1. (sqrt{10m})
  2. (sqrt[5]{3 n})
  3. (3 sqrt[4]{6 y})
Answer
  1. ((10 m)^{frac{1}{2}})
  2. ((3 n)^{frac{1}{5}})
  3. (3(6 y)^{frac{1}{4}})

Exercise (PageIndex{4})

Write with a rational exponent:

  1. (sqrt[7]{3 k})
  2. (sqrt[4]{5 j})
  3. (8 sqrt[3]{2 a})
Answer
  1. ((3 k)^{frac{1}{7}})
  2. ((5 j)^{frac{1}{4}})
  3. (8(2 a)^{frac{1}{3}})

In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.

Example (PageIndex{3})

Simplify:

  1. (25^{frac{1}{2}})
  2. (64^{frac{1}{3}})
  3. (256^{frac{1}{4}})

Solution:

a.

(25^{frac{1}{2}})

Rewrite as a square root.

(sqrt{25})

Simplify.

(5)

b.

(64^{frac{1}{3}})

Rewrite as a cube root.

(sqrt[3]{64})

Recognize (64) is a perfect cube.

(sqrt[3]{4^{3}})

Simplify.

(4)

c.

(256^{frac{1}{4}})

Rewrite as a fourth root.

(sqrt[4]{256})

Recognize (256) is a perfect fourth power.

(sqrt[4]{4^{4}})

Simplify.

(4)

Exercise (PageIndex{5})

Simplify:

  1. (36^{frac{1}{2}})
  2. (8^{frac{1}{3}})
  3. (16^{frac{1}{4}})
Answer
  1. (6)
  2. (2)
  3. (2)

Exercise (PageIndex{6})

Simplify:

  1. (100^{frac{1}{2}})
  2. (27^{frac{1}{3}})
  3. (81^{frac{1}{4}})
Answer
  1. (10)
  2. (3)
  3. (3)

Be careful of the placement of the negative signs in the next example. We will need to use the property (a^{-n}=frac{1}{a^{n}}) in one case.

Example (PageIndex{4})

Simplify:

  1. ((-16)^{frac{1}{4}})
  2. (-16^{frac{1}{4}})
  3. ((16)^{-frac{1}{4}})

Solution:

a.

((-16)^{frac{1}{4}})

Rewrite as a fourth root.

(sqrt[4]{-16})

(sqrt[4]{(-2)^{4}})

Simplify.

No real solution

b.

(-16^{frac{1}{4}})

The exponent only applies to the (16). Rewrite as a fourth root.

(-sqrt[4]{16})

Rewrite (16) as (2^{4})

(-sqrt[4]{2^{4}})

Simplify.

(-2)

c.

((16)^{-frac{1}{4}})

Rewrite using the property (a^{-n}=frac{1}{a^{n}}).

(frac{1}{(16)^{frac{1}{4}}})

Rewrite as a fourth root.

(frac{1}{sqrt[4]{16}})

Rewrite (16) as (2^{4}).

(frac{1}{sqrt[4]{2^{4}}})

Simplify.

(frac{1}{2})

Exercise (PageIndex{7})

Simplify:

  1. ((-64)^{-frac{1}{2}})
  2. (-64^{frac{1}{2}})
  3. ((64)^{-frac{1}{2}})
Answer
  1. No real solution
  2. (-8)
  3. (frac{1}{8})

Exercise (PageIndex{8})

Simplify:

  1. ((-256)^{frac{1}{4}})
  2. (-256^{frac{1}{4}})
  3. ((256)^{-frac{1}{4}})
Answer
  1. No real solution
  2. (-4)
  3. (frac{1}{4})

Simplify Expressions with (a^{frac{m}{n}})

We can look at (a^{frac{m}{n}}) in two ways. Remember the Power Property tells us to multiply the exponents and so (left(a^{frac{1}{n}} ight)^{m}) and (left(a^{m} ight)^{frac{1}{n}}) both equal (a^{frac{m}{n}}). If we write these expressions in radical form, we get

(a^{frac{m}{n}}=left(a^{frac{1}{n}} ight)^{m}=(sqrt[n]{a})^{m} quad ext { and } quad a^{frac{m}{n}}=left(a^{m} ight)^{^{frac{1}{n}}}=sqrt[n]{a^{m}})

This leads us to the following defintion.

Definition (PageIndex{2}): Rational Exponent (a^{frac{m}{n}})

For any positive integers (m) and (n),

(a^{frac{m}{n}}=(sqrt[n]{a})^{m} quad ext { and } quad a^{frac{m}{n}}=sqrt[n]{a^{m}})

Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller, before raising it to the power indicated.

Example (PageIndex{5})

Write with a rational exponent:

  1. (sqrt{y^{3}})
  2. ((sqrt[3]{2 x})^{4})
  3. (sqrt{left(frac{3 a}{4 b} ight)^{3}})

Solution:

We want to use (a^{frac{m}{n}}=sqrt[n]{a^{m}}) to write each radical in the form (a^{frac{m}{n}})

a.

b.

c.

Exercise (PageIndex{9})

Write with a rational exponent:

  1. (sqrt{x^{5}})
  2. ((sqrt[4]{3 y})^{3})
  3. (sqrt{left(frac{2 m}{3 n} ight)^{5}})
Answer
  1. (x^{frac{5}{2}})
  2. ((3 y)^{frac{3}{4}})
  3. (left(frac{2 m}{3 n} ight)^{frac{5}{2}})

Exercise (PageIndex{10})

Write with a rational exponent:

  1. (sqrt[5]{a^{2}})
  2. ((sqrt[3]{5 a b})^{5})
  3. (sqrt{left(frac{7 x y}{z} ight)^{3}})
Answer
  1. (a^{frac{2}{5}})
  2. ((5 a b)^{frac{5}{3}})
  3. (left(frac{7 x y}{z} ight)^{frac{3}{2}})

Remember that (a^{-n}=frac{1}{a^{n}}). The negative sign in the exponent does not change the sign of the expression.

Example (PageIndex{6})

Simplify:

  1. (125^{frac{2}{3}})
  2. (16^{-frac{3}{2}})
  3. (32^{-frac{2}{5}})

Solution:

We will rewrite the expression as a radical first using the defintion, (a^{frac{m}{n}}=(sqrt[n]{a})^{m}). This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.

a.

(125^{frac{2}{3}})

The power of the radical is the numerator of the exponent, (2). The index of the radical is the denominator of the exponent, (3).

((sqrt[3]{125})^{2})

Simplify.

((5)^{2})

(25)

b. We will rewrite each expression first using (a^{-n}=frac{1}{a^{n}}) and then change to radical form.

(16^{-frac{3}{2}})

Rewrite using (a^{-n}=frac{1}{a^{n}})

(frac{1}{16^{frac{3}{2}}})

Change to radical form. The power of the radical is the numerator of the exponent, (3). The index is the denominator of the exponent, (2).

(frac{1}{(sqrt{16})^{3}})

Simplify.

(frac{1}{4^{3}})

(frac{1}{64})

c.

(32^{-frac{2}{5}})

Rewrite using (a^{-n}=frac{1}{a^{n}})

(frac{1}{32^{frac{2}{5}}})

Change to radical form.

(frac{1}{(sqrt[5]{32})^{2}})

Rewrite the radicand as a power.

(frac{1}{left(sqrt[5]{2^{5}} ight)^{2}})

Simplify.

(frac{1}{2^{2}})

(frac{1}{4})

Exercise (PageIndex{11})

Simplify:

  1. (27^{frac{2}{3}})
  2. (81^{-frac{3}{2}})
  3. (16^{-frac{3}{4}})
Answer
  1. (9)
  2. (frac{1}{729})
  3. (frac{1}{8})

Exercise (PageIndex{12})

Simplify:

  1. (4^{frac{3}{2}})
  2. (27^{-frac{2}{3}})
  3. (625^{-frac{3}{4}})
Answer
  1. (8)
  2. (frac{1}{9})
  3. (frac{1}{125})

Example (PageIndex{7})

Simplify:

  1. (-25^{frac{3}{2}})
  2. (-25^{-frac{3}{2}})
  3. ((-25)^{frac{3}{2}})

Solution:

a.

(-25^{frac{3}{2}})

Rewrite in radical form.

(-(sqrt{25})^{3})

Simplify the radical.

(-(5)^{3})

Simplify.

(-125)

b.

(-25^{-frac{3}{2}})

Rewrite using (a^{-n}=frac{1}{a^{n}}).

(-left(frac{1}{25^{frac{3}{2}}} ight))

Rewrite in radical form.

(-left(frac{1}{(sqrt{25})^{3}} ight))

Simplify the radical.

(-left(frac{1}{(5)^{3}} ight))

Simplify.

(-frac{1}{125})

c.

((-25)^{frac{3}{2}})

Rewrite in radical form.

((sqrt{-25})^{3})

There is no real number whose square root is (-25).

Not a real number.

Exercise (PageIndex{13})

Simplify:

  1. (-16^{frac{3}{2}})
  2. (-16^{-frac{3}{2}})
  3. ((-16)^{-frac{3}{2}})
Answer
  1. (-64)
  2. (-frac{1}{64})
  3. Not a real number

Exercise (PageIndex{14})

Simplify:

  1. (-81^{frac{3}{2}})
  2. (-81^{-frac{3}{2}})
  3. ((-81)^{-frac{3}{2}})
Answer
  1. (-729)
  2. (-frac{1}{729})
  3. Not a real number

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponents here to have them for reference as we simplify expressions.

Properties of Exponents

If (a) and (b) are real numbers and (m) and (n) are rational numbers, then

Product Property

(a^{m} cdot a^{n}=a^{m+n})

Power Property

(left(a^{m} ight)^{n}=a^{m cdot n})

Product to a Power

((a b)^{m}=a^{m} b^{m})

Quotient Property

(frac{a^{m}}{a^{n}}=a^{m-n}, a eq 0)

Zero Exponent Definition

(a^{0}=1, a eq 0)

Quotient to a Power Property

(left(frac{a}{b} ight)^{m}=frac{a^{m}}{b^{m}}, b eq 0)

Negative Exponent Property

(a^{-n}=frac{1}{a^{n}}, a eq 0)

We will apply these properties in the next example.

Example (PageIndex{8})

Simplify:

  1. (x^{frac{1}{2}} cdot x^{frac{5}{6}})
  2. (left(z^{9} ight)^{frac{2}{3}})
  3. (frac{x^{frac{1}{3}}}{x^{frac{5}{3}}})

Solution

a. The Product Property tells us that when we multiple the same base, we add the exponents.

(x^{frac{1}{2}} cdot x^{frac{5}{6}})

The bases are the same, so we add the exponents.

(x^{frac{1}{2}+frac{5}{6}})

Add the fractions.

(x^{frac{8}{6}})

Simplify the exponent.

(x^{frac{4}{3}})

b. The Power Property tells us that when we raise a power to a power, we multiple the exponents.

(left(z^{9} ight)^{frac{2}{3}})

To raise a power to a power, we multiple the exponents.

(z^{9 cdot frac{2}{3}})

Simplify.

(z^{6})

c. The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

(frac{x^{frac{1}{3}}}{x^{frac{5}{3}}})

To divide with the same base, we subtract the exponents.

(frac{1}{x^{frac{5}{3}-frac{1}{3}}})

Simplify.

(frac{1}{x^{frac{4}{3}}})

Exercise (PageIndex{15})

Simplify:

  1. (x^{frac{1}{6}} cdot x^{frac{4}{3}})
  2. (left(x^{6} ight)^{frac{4}{3}})
  3. (frac{x^{frac{2}{3}}}{x^{frac{5}{3}}})
Answer
  1. (x^{frac{3}{2}})
  2. (x^{8})
  3. (frac{1}{x})

Exercise (PageIndex{16})

Simplify:

  1. (y^{frac{3}{4}} cdot y^{frac{5}{8}})
  2. (left(m^{9} ight)^{frac{2}{9}})
  3. (frac{d^{frac{1}{5}}}{d^{frac{6}{5}}})
Answer
  1. (y^{frac{11}{8}})
  2. (m^{2})
  3. (frac{1}{d})

Sometimes we need to use more than one property. In the next example, we will use both the Product to a Power Property and then the Power Property.

Example (PageIndex{9})

Simplify:

  1. (left(27 u^{frac{1}{2}} ight)^{frac{2}{3}})
  2. (left(m^{frac{2}{3}} n^{frac{1}{2}} ight)^{frac{3}{2}})

Solution:

a.

(left(27 u^{frac{1}{2}} ight)^{frac{2}{3}})

First we use the Product to a Power Property.

((27)^{frac{2}{3}}left(u^{frac{1}{2}} ight)^{frac{2}{3}})

Rewrite (27) as a power of (3).

(left(3^{3} ight)^{frac{2}{3}}left(u^{frac{1}{2}} ight)^{frac{2}{3}})

To raise a power to a power, we multiple the exponents.

(left(3^{2} ight)left(u^{frac{1}{3}} ight))

Simplify.

(9 u^{frac{1}{3}})

b.

(left(m^{frac{2}{3}} n^{frac{1}{2}} ight)^{frac{3}{2}})

First we use the Product to a Power Property.

(left(m^{frac{2}{3}} ight)^{frac{3}{2}}left(n^{frac{1}{2}} ight)^{frac{3}{2}})

To raise a power to a power, we multiply the exponents.

(m n^{frac{3}{4}})

Exercise (PageIndex{17})

Simplify:

  1. (left(32 x^{frac{1}{3}} ight)^{frac{3}{5}})
  2. (left(x^{frac{3}{4}} y^{frac{1}{2}} ight)^{frac{2}{3}})
Answer
  1. (8 x^{frac{1}{5}})
  2. (x^{frac{1}{2}} y^{frac{1}{3}})

Exercise (PageIndex{18})

Simplify:

  1. (left(81 n^{frac{2}{5}} ight)^{frac{3}{2}})
  2. (left(a^{frac{3}{2}} b^{frac{1}{2}} ight)^{frac{4}{3}})
Answer
  1. (729 n^{frac{3}{5}})
  2. (a^{2} b^{frac{2}{3}})

We will use both the Product Property and the Quotient Property in the next example.

Example (PageIndex{10})

Simplify:

  1. (frac{x^{frac{3}{4}} cdot x^{-frac{1}{4}}}{x^{-frac{6}{4}}})
  2. (left(frac{16 x^{frac{4}{3}} y^{-frac{5}{6}}}{x^{-frac{2}{3}} y^{frac{1}{6}}} ight)^{frac{1}{2}})

Solution:

a.

(frac{x^{frac{3}{4}} cdot x^{-frac{1}{4}}}{x^{-frac{6}{4}}})

Use the Product Property in the numerator, add the exponents.

(frac{x^{frac{2}{4}}}{x^{-frac{6}{4}}})

Use the Quotient Property, subtract the exponents.

(x^{frac{8}{4}})

Simplify.

(x^{2})

b.

(left(frac{16 x^{frac{4}{3}} y^{-frac{5}{6}}}{x^{-frac{2}{3}} y^{frac{1}{6}}} ight)^{frac{1}{2}})

Use the Quotient Property, subtract the exponents.

(left(frac{16 x^{frac{6}{3}}}{y^{frac{6}{6}}} ight)^{frac{1}{2}})

Simplify.

(left(frac{16 x^{2}}{y} ight)^{frac{1}{2}})

Use the Product to a Power Property, multiply the exponents.

(frac{4 x}{y^{frac{1}{2}}})

Exercise (PageIndex{19})

Simplify:

  1. (frac{m^{frac{2}{3}} cdot m^{-frac{1}{3}}}{m^{-frac{5}{3}}})
  2. (left(frac{25 m^{frac{1}{6}} n^{frac{11}{6}}}{m^{frac{2}{3}} n^{-frac{1}{6}}} ight)^{frac{1}{2}})
Answer
  1. (m^{2})
  2. (frac{5 n}{m^{frac{1}{4}}})

Exercise (PageIndex{20})

Simplify:

  1. (frac{u^{frac{4}{5}} cdot u^{-frac{2}{5}}}{u^{-frac{13}{5}}})
  2. (left(frac{27 x^{frac{4}{5}} y^{frac{1}{6}}}{x^{frac{1}{5}} y^{-frac{5}{6}}} ight)^{frac{1}{3}})
Answer
  1. (u^{3})
  2. (3 x^{frac{1}{5}} y^{frac{1}{3}})

Access these online resources for additional instruction and practice with simplifying rational exponents.

  • Review-Rational Exponents
  • Using Laws of Exponents on Radicals: Properties of Rational Exponents

Key Concepts

  • Rational Exponent (a^{frac{1}{n}})
    • If (sqrt[n]{a}) is a real number and (n≥2), then (a^{frac{1}{n}}=sqrt[n]{a}).
  • Rational Exponent (a^{frac{m}{n}})
    • For any positive integers (m) and (n),
      (a^{frac{m}{n}}=(sqrt[n]{a})^{m} ext { and } a^{frac{m}{n}}=sqrt[n]{a^{m}})
  • Properties of Exponents
    • If (a, b) are real numbers and (m, n) are rational numbers, then
      • Product Property (a^{m} cdot a^{n}=a^{m+n})
      • Power Property (left(a^{m} ight)^{n}=a^{m cdot n})
      • Product to a Power ((a b)^{m}=a^{m} b^{m})
      • Quotient Property (frac{a^{m}}{a^{n}}=a^{m-n}, a eq 0)
      • Zero Exponent Definition (a^{0}=1, a eq 0)
      • Quotient to a Power Property (left(frac{a}{b} ight)^{m}=frac{a^{m}}{b^{m}}, b eq 0)
      • Negative Exponent Property (a^{-n}=frac{1}{a^{n}}, a eq 0)

Simplifying Polynomials

In section 3 of chapter 1 there are several very important definitions, which we have used many times. Since these definitions take on new importance in this chapter, we will repeat them.

When an algebraic expression is composed of parts connected by + or - signs, these parts, along with their signs, are called the terms of the expression.

a + b has two terms.
2x + 5y - 3 has three terms.

In a + b the terms are a and b. In 2x + 5y - 3 the terms are 2x, 5y, and -3.

When an algebraic expression is composed of parts to be multiplied, these parts are called the factors of the expression.

It is very important to be able to distinguish between terms and factors. Rules that apply to terms will not, in general, apply to factors. When naming terms or factors, it is necessary to regard the entire expression.

From now on through all algebra you will be using the words term and factor. Make sure you understand the definitions.

An exponent is a numeral used to indicate how many times a factor is to be used in a product. An exponent is usually written as a smaller (in size) numeral slightly above and to the right of the factor affected by the exponent.

An exponent is sometimes referred to as a "power." For example, 5 3 could be referred to as "five to the third power."

Note the difference between 2x 3 and (2x) 3 . From using parentheses as grouping symbols we see that

2x 3 means 2(x)(x)(x), whereas (2x) 3 means (2x)(2x)(2x) or 8x 3 .

Unless parentheses are used, the exponent only affects the factor directly preceding it.

In an expression such as 5x 4
5 is the coefficient,
x is the base,
4 is the exponent.
5x 4 means 5(x)(x)(x)(x).

Note that only the base is affected by the exponent.

Many students make the error of multiplying the base by the exponent.For example, they will say 3 4 = 12 instead of the correct answer,
3 4 = (3)(3)(3)(3) = 81.

When we write a literal number such as x, it will be understood that the coefficient is one and the exponent is one. This can be very important in many operations.

It is also understood that a written numeral such as 3 has an exponent of 1. We just do not bother to write an exponent of 1.


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Simplify the given expression to rational exponent form and justify each step by identifying the properties of rational exponents used. All work must be shown. 1/cubed root of x to -6 exponent

Math 9

use exponent rules to simplify. Write as a single power, don't need to find the value. [ (11/20)^4 x (11/20)^-8]^5 and (11/20)^-6 you cant leave an number to the power of a negative exponent. You have to change it like this ex :


4.3: Simplify Rational Exponents

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4.3: Simplify Rational Exponents

· Convert radicals to expressions with rational exponents.

· Convert expressions with rational exponents to their radical equivalent.

· Use the laws of exponents to simplify expressions with rational exponents.

· Use rational exponents to simplify radical expressions.

Square roots are most often written using a radical sign, like this, . But there is another way to represent the taking of a root. You can use rational exponents instead of a radical. A rational exponent is an exponent that is a fraction. For example, can be written as .

Can’t imagine raising a number to a rational exponent? They may be hard to get used to, but rational exponents can actually help simplify some problems. Let’s explore the relationship between rational (fractional) exponents and radicals.

Rewriting Radical Expressions Using Rational Exponents

Radicals and fractional exponents are alternate ways of expressing the same thing. You have already seen how square roots can be expressed as an exponent to the power of one-half.


4.3: Simplify Rational Exponents

Perform the indicated operations.
Write each answer using only positive exponents.
Assume all variables represent nonzero real numbers.

Perform the indicated operations.
Write each answer using only positive exponents.
Assume all variables represent nonzero real numbers.

Perform the indicated operations.
Write each answer using only positive exponents.
Assume all variables represent nonzero real numbers.

Evaluate the expression   16 1 &frasl4 = ( 2 4 ) 1 &frasl4
Evaluate the expression   16 1 &frasl4 = 2

Perform the indicated operations.
Write each answer using only positive exponents.
Assume all variables represent positive real numbers.

100 3 &frasl2 = ( 10 2 ) 3 &frasl2
100 3 &frasl2 = 10 3
100 3 &frasl2 = 1000

Perform the indicated operations.
Write each answer using only positive exponents.
Assume all variables represent positive real numbers.

Perform the indicated operations.
Write each answer using only positive exponents.
Assume all variables represent positive real numbers.

Find the product.
Assume all variables represent positive real numbers.

Factor, using the given common factor.
Assume all variables represent positive real numbers.

4t מ + 8t נ ,   given 4t נ

4t מ + 8t נ = 4t נ ( t 2 + 2 )

Factor, using the given common factor.
Assume all variables represent positive real numbers.

( p + 4 ) ן &frasl2 + ( p + 4 ) ם &frasl2 + ( p + 4 ) 1 &frasl2 = ( p + 4 ) ן &frasl2 [ 1 + ( p + 4 ) + ( p + 4 ) 2 ]
( p + 4 ) ן &frasl2 + ( p + 4 ) ם &frasl2 + ( p + 4 ) 1 &frasl2 = ( p + 4 ) ן &frasl2 [ 1 + ( p + 4 ) + ( p 2 + 8p + 16 ) ]
( p + 4 ) ן &frasl2 + ( p + 4 ) ם &frasl2 + ( p + 4 ) 1 &frasl2 = ( p + 4 ) ן &frasl2 [ 1 + p + 4 + p 2 + 8p + 16 ]
( p + 4 ) ן &frasl2 + ( p + 4 ) ם &frasl2 + ( p + 4 ) 1 &frasl2 = ( p + 4 ) ן &frasl2 [ p 2 + 9p + 21 ]

Perform all indicated operations and write the answer with positive integer exponents.

Simplify the rational expression.
Use factoring as needed.
Assume all variable expressions represent positive real numbers.

2 ( 2x – 3 ) 1 &frasl3 – ( x – 1 ) ( 2x – 3 ) מ &frasl3
–––––––––––––––––––––––––––––––
( 2x – 3 ) 2 &frasl3
 =  ( 2x – 3 ) 1 &frasl3 [ 2 – ( x – 1 ) ( 2x – 3 ) ם ]
–––––––––––––––––––––––––––––––––
( 2x – 3 ) 1 &frasl3 ( 2x – 3 ) 1 &frasl3

2 ( 2x – 3 ) 1 &frasl3 – ( x – 1 ) ( 2x – 3 ) מ &frasl3
–––––––––––––––––––––––––––––––
( 2x – 3 ) 2 &frasl3
 =  2 – ( x – 1 ) ( 2x – 3 ) ם
––––––––––––––––––––
( 2x – 3 ) 1 &frasl3

2 ( 2x – 3 ) 1 &frasl3 – ( x – 1 ) ( 2x – 3 ) מ &frasl3
–––––––––––––––––––––––––––––––
( 2x – 3 ) 2 &frasl3

 = 
2 –  x – 1
––––––
2x – 3
–––––––––––
( 2x – 3 ) 1 &frasl3

2 ( 2x – 3 ) 1 &frasl3 – ( x – 1 ) ( 2x – 3 ) מ &frasl3
–––––––––––––––––––––––––––––––
( 2x – 3 ) 2 &frasl3

Source of exercise problems:   College Algebra and Trigonometry by Lial, Hornsey, Schneider, Daniels, Fifth Edition, Section R6, pp. 55-58


4.3: Simplify Rational Exponents

Now that we have looked at integer exponents we need to start looking at more complicated exponents. In this section we are going to be looking at rational exponents. That is exponents in the form

where both (m) and (n) are integers.

We will start simple by looking at the following special case,

where (n) is an integer. Once we have this figured out the more general case given above will actually be pretty easy to deal with.

Let’s first define just what we mean by exponents of this form.

In other words, when evaluating (>>) we are really asking what number (in this case (a)) did we raise to the (n) to get (b). Often (>>) is called the (n) th root of b.

Let’s do a couple of evaluations.

When doing these evaluations, we will not actually do them directly. When first confronted with these kinds of evaluations doing them directly is often very difficult. In order to evaluate these we will remember the equivalence given in the definition and use that instead.

We will work the first one in detail and then not put as much detail into the rest of the problems.

So, here is what we are asking in this problem.

Using the equivalence from the definition we can rewrite this as,

So, all that we are really asking here is what number did we square to get 25. In this case that is (hopefully) easy to get. We square 5 to get 25. Therefore,

So what we are asking here is what number did we raise to the 5 th power to get 32?

What number did we raise to the 4 th power to get 81?

We need to be a little careful with minus signs here, but other than that it works the same way as the previous parts. What number did we raise to the 3 rd power (i.e. cube) to get -8?

This part does not have an answer. It is here to make a point. In this case we are asking what number do we raise to the 4 th power to get -16. However, we also know that raising any number (positive or negative) to an even power will be positive. In other words, there is no real number that we can raise to the 4 th power to get -16.

Note that this is different from the previous part. If we raise a negative number to an odd power we will get a negative number so we could do the evaluation in the previous part.

As this part has shown, we can’t always do these evaluations.

Again, this part is here to make a point more than anything. Unlike the previous part this one has an answer. Recall from the previous section that if there aren’t any parentheses then only the part immediately to the left of the exponent gets the exponent. So, this part is really asking us to evaluate the following term.

So, we need to determine what number raised to the 4 th power will give us 16. This is 2 and so in this case the answer is,

As the last two parts of the previous example has once again shown, we really need to be careful with parenthesis. In this case parenthesis makes the difference between being able to get an answer or not.

Also, don’t be worried if you didn’t know some of these powers off the top of your head. They are usually fairly simple to determine if you don’t know them right away. For instance, in the part b we needed to determine what number raised to the 5 will give 32. If you can’t see the power right off the top of your head simply start taking powers until you find the correct one. In other words compute (<2^5>), (<3^5>), (<4^5>) until you reach the correct value. Of course, in this case we wouldn’t need to go past the first computation.

The next thing that we should acknowledge is that all of the properties for exponents that we gave in the previous section are still valid for all rational exponents. This includes the more general rational exponent that we haven’t looked at yet.

Now that we know that the properties are still valid we can see how to deal with the more general rational exponent. There are in fact two different ways of dealing with them as we’ll see. Both methods involve using property 2 from the previous section. For reference purposes this property is,

So, let’s see how to deal with a general rational exponent. We will first rewrite the exponent as follows.

In other words, we can think of the exponent as a product of two numbers. Now we will use the exponent property shown above. However, we will be using it in the opposite direction than what we did in the previous section. Also, there are two ways to do it. Here they are,

Using either of these forms we can now evaluate some more complicated expressions

We can use either form to do the evaluations. However, it is usually more convenient to use the first form as we will see.

Let’s use both forms here since neither one is too bad in this case. Let’s take a look at the first form.

Now, let’s take a look at the second form.

So, we get the same answer regardless of the form. Notice however that when we used the second form we ended up taking the 3 rd root of a much larger number which can cause problems on occasion.

Again, let’s use both forms to compute this one.

As this part has shown the second form can be quite difficult to use in computations. The root in this case was not an obvious root and not particularly easy to get if you didn’t know it right off the top of your head.

In this case we’ll only use the first form. However, before doing that we’ll need to first use property 5 of our exponent properties to get the exponent onto the numerator and denominator.

We can also do some of the simplification type problems with rational exponents that we saw in the previous section.

For this problem we will first move the exponent into the parenthesis then we will eliminate the negative exponent as we did in the previous section. We will then move the term to the denominator and drop the minus sign.

In this case we will first simplify the expression inside the parenthesis.

Don’t worry if, after simplification, we don’t have a fraction anymore. That will happen on occasion. Now we will eliminate the negative in the exponent using property 7 and then we’ll use property 4 to finish the problem up.

We will leave this section with a warning about a common mistake that students make in regard to negative exponents and rational exponents. Be careful not to confuse the two as they are totally separate topics.


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Watch the video: Lesson Simplifying Rational Expressions (October 2021).