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5. Rational Exponents & Radical Functions
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United States of AmericaTennessee
Algebra 2

5.02 Properties of rational exponents

Lesson
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Lesson

Concept summary

Expressions with rational exponents are expressions where the exponent is a rational number (can be written as an integer fraction). In general, a rational exponent can be rewritten as a radical (or a radical as a rational exponent) in the following ways:

a^\frac{m}{n}=\sqrt[n]{a^m}, \text{ where } a>0 \\a^\frac{m}{n}=\left(\sqrt[n]{a}\right)^m, \text{ where } a>0

The laws of exponents can also be applied to expressions with rational exponents, where m,n,p and q are integers and a and b are nonzero real numbers

  • Product of powers a^\frac{m}{n} \cdot a^\frac{p}{n} = a^\frac{m+p}{n}
  • Quotient of powers \dfrac{{a}^\frac{m}{n}}{{a}^\frac{p}{n}}=a^\frac{m-p}{n}
  • Power of a power (a^\frac{m}{n})^\frac{p}{q} = a^\frac{mp}{nq}
  • Power of a product (a b)^\frac{m}{n} = a^\frac{m}{n} \cdot b^\frac{m}{n}
  • Identity exponent a^1=a
  • Zero exponent a^0=1
  • Negative exponent a^{- \frac{m}{n}}=\frac{1}{a^\frac{m}{n}}

Worked examples

Example 1

Write each of the following expressions in exponential form. Assume all variables are positive.

a

\sqrt[5]{x^7}

Approach

We can write this expression in exponential form using the fact \sqrt[n]{a^m}=a^{\frac{m}{n}}

Solution

\sqrt[5]{x^7}=x^{\frac{5}{7}}

b

\left(\sqrt[4]{x^3y^5}\right)^{12}

Approach

We can use a number of power laws to help us solve this. We can first use the property of rational exponents to rewrite the radical with a rational exponent. We can then use the power of a power law to simplify the powers.

Solution

\displaystyle \left(\sqrt[4]{x^3y^5}\right)^{12} \displaystyle =\displaystyle \left(\left(x^3y^5\right)^{\frac{1}{4}}\right)^{12}Rational exponent law
\displaystyle =\displaystyle \left(x^3y^5\right)^{\frac{1}{4}\left(12\right)}Power of a power
\displaystyle =\displaystyle \left(x^3y^5\right)^{3}Simplify the exponent
\displaystyle =\displaystyle \left(x^3\right)^3\left(y^5\right)^3Power of a product
\displaystyle =\displaystyle x^9y^{15}Power of a power

Example 2

Write each of the following expressions in reduced radical form. Assume all variables are positive.

a

\left(bc\right)^{\frac{1}{5}}

Approach

We can write this expression in exponential form using the fact a^{\frac{m}{n}}=\sqrt[n]{a^m}

Solution

\left(bc\right)^{\frac{1}{5}}=\sqrt[5]{bc}

b

\left(\dfrac{81x}{5y^8}\right)^{\frac{1}{4}}

Approach

We first want to rewrite the expression as a radical and simplify.

Solution

\displaystyle \left(\dfrac{81x}{5y^8}\right)^{\frac{1}{4}}\displaystyle =\displaystyle \sqrt[4]{\dfrac{81x}{5y^8}}Rational exponent law
\displaystyle =\displaystyle \frac{\sqrt[4]{81x}}{\sqrt[4]{5y^8}}Rewrite using properties of radicals
\displaystyle =\displaystyle \frac{3\sqrt[4]{x}}{\sqrt[4]{5}y^2}Simplify the radical

Example 3

Fully simplify each of the following expressions. Write the results in exponential form. Assume all variables are positive.

a

2c^{\frac{1}{5}}\cdot 3c^{\frac{2}{5}}

Approach

We have a product of powers with rational exponents, so we can simplify using the product of powers law:a^\frac{m}{n} \cdot a^\frac{p}{n} = a^\frac{m+p}{n} As the denominators are the same, we can add the numerators.

Solution

\displaystyle 2c^{\frac{1}{5}}\cdot3c^{\frac{2}{5}}\displaystyle =\displaystyle 2\left(3\right)c^{\frac{1}{5}}c^{\frac{2}{5}}Product of coefficients
\displaystyle =\displaystyle 2\left(3\right)c^{\frac{1}{5}+\frac{2}{5}}Product of powers
\displaystyle =\displaystyle 6c^{\frac{3}{5}}Simplify the exponent
b

\dfrac{15 p^{\frac{7}{3}}}{5 p^{\frac{1}{6}}}

Approach

We have a quotient of powers with rational exponents, so we can simplify using the quotient of powers law:\dfrac{a^\frac{m}{n}} {a^\frac{p}{n}} = a^\frac{m-p}{n} As the denominators are different, we will need to express the rational exponents with common denominators. We can also simplify the quotient of the numeric coefficients.

Solution

\displaystyle \dfrac{15 p^{\frac{7}{3}}}{5 p^{\frac{1}{6}}}\displaystyle =\displaystyle 3\dfrac{ p^{\frac{7}{3}}}{ p^{\frac{1}{6}}}Simplify the numeric coefficients
\displaystyle =\displaystyle 3p^{\frac{7}{3}-\frac{1}{6}}Quotient of powers
\displaystyle =\displaystyle 3p^{\frac{14}{6}-\frac{1}{6}}Express with common denominator
\displaystyle =\displaystyle 3p^{\frac{13}{6}}Simplify the rational exponent

Example 4

Fully simplify the following expressions. Write the result in reduced radical form. Assume all variables are positive.\left(2a^{\frac{1}{4}}\right)\left(16a\right)^{\frac{1}{2}}

Approach

We can first evaluate the rational exponent that is applied to the entire second term using the power of a product law. We can then use the product of powers law to combine the two rational exponents. Lastly, we want to rewrite the expression in reduced radical form.

Solution

\displaystyle \left(2a^{\frac{1}{4}}\right)\left(16a\right)^{\frac{1}{2}}\displaystyle =\displaystyle 2a^{\frac{1}{4}}4a^{\frac{1}{2}}Power of a product
\displaystyle =\displaystyle 8a^{\frac{1}{4}}a^{\frac{1}{2}}Product of coefficients
\displaystyle =\displaystyle 8a^{\frac{1}{4}+\frac{1}{2}}Product of powers
\displaystyle =\displaystyle 8a^{\frac{1}{4}+\frac{2}{4}}Express with common denominator
\displaystyle =\displaystyle 8a^{\frac{3}{4}}Simplify the rational exponent
\displaystyle =\displaystyle 8\sqrt[4]{a^3}Express in radical form

Outcomes

A2.N.RN.A.1

Extend the properties of integer exponents to rational exponents.

A2.N.RN.A.1.A

Develop the meaning of rational exponents by applying the properties of integer exponents.

A2.N.RN.A.1.B

Explain why x^(1/n) can be written as the nth root of x.

A2.N.RN.A.1.C

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

A2.MP1

Make sense of problems and persevere in solving them.

A2.MP6

Attend to precision.

A2.MP7

Look for and make use of structure.

A2.MP8

Look for and express regularity in repeated reasoning.

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