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5. Exponential Functions
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United States of AmericaFL
Grade 912

5.02 Rational exponents

Lesson
Practice
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

For each expression, convert between exponential and radical forms.

a

8^{\frac{2}{5}}

Approach

We can use a^\frac{m}{n}=\sqrt[n]{a^m} to first rewrite the expression, then check to see if the radical can be simplified.

Solution

\displaystyle 8^{\frac{2}{5}}\displaystyle =\displaystyle \sqrt[5]{8^2}
\displaystyle =\displaystyle \sqrt[5]{64}
b

\sqrt[4]{33}

Approach

We can use a^\frac{m}{n}=\sqrt[n]{a^m} to rewrite the expression.

Solution

\sqrt[4]{33} = 33^\frac{1}{4}

Example 2

Rewrite the following as an expression with a single exponent:

a

3^\frac{7}{2} \cdot 3^\frac{9}{2}

Approach

We can use the product of powers, a^\frac{m}{n} \cdot a^\frac{p}{n} = a^\frac{m+p}{n} to rewrite the expression.

Solution

\displaystyle 3^\frac{7}{2} \cdot 3^\frac{9}{2}\displaystyle =\displaystyle 3^\frac{7+9}{2}Product of powers law
\displaystyle =\displaystyle 3^\frac{16}{2}Evaluate the addition
\displaystyle =\displaystyle 3^8Evaluate the division
b

5^\frac{8}{3} \cdot 2^\frac{8}{3}

Approach

We can use the power of a product (ab)^\frac{m}{n} = a^\frac{m}{n} \cdot b^\frac{m}{n} to rewrite the expression.

Solution

\displaystyle 5^\frac{8}{3} \cdot 2^\frac{8}{3}\displaystyle =\displaystyle (5 \cdot 2)^\frac{8}{3}Power of a product law
\displaystyle =\displaystyle 10^\frac{8}{3}Evaluate the multiplication
c

(2^\frac{3}{5})^\frac{4}{3}

Approach

We can use the power of a power (a^\frac{m}{n})^\frac{p}{q} = a^\frac{m \cdot p}{n \cdot q} to rewrite the expression.

Solution

\displaystyle (2^\frac{3}{5})^\frac{4}{3}\displaystyle =\displaystyle 2^{\frac{3 \cdot 4}{5 \cdot 3}}Power of a power law
\displaystyle =\displaystyle 2^{\frac{12}{15}}Evaluate the multiplication
\displaystyle =\displaystyle 2^{\frac{4}{5}}Simplify the fraction
d

\dfrac{15^\frac{32}{11}}{15^\frac{21}{11}}

Approach

We can use the quotient of powers \dfrac{{a}^\frac{m}{n}}{{a}^\frac{p}{n}}=a^\frac{m-p}{n} to rewrite the expression.

Solution

\displaystyle \dfrac{15^\frac{32}{11}}{15^\frac{21}{11}}\displaystyle =\displaystyle 15^\frac{32-21}{11}Quotient of powers law
\displaystyle =\displaystyle 15^\frac{11}{11}Evaluate the subtraction
\displaystyle =\displaystyle 15^1Evaluate the division
\displaystyle =\displaystyle 15Identity exponent

Example 3

Evaluate the following expressions:

a

64^{ - \frac{1}{6}}

Approach

We can use the definition of negative exponents, a^{- \frac{m}{n}}=\frac{1}{a^\frac{m}{n}}, to evaluate the expression.

Solution

\displaystyle 64^{ - \frac{1}{6}}\displaystyle =\displaystyle \frac{1}{{64}^\frac{1}{6}}Definition of negative exponents
\displaystyle =\displaystyle \frac{1}{\sqrt[6]{64}}Definition of rational exponents
\displaystyle =\displaystyle \frac{1}{2}Evaluate the radical
b

\dfrac{4^{- \frac{2}{9}}}{4^{- \frac{2}{9}}}

Approach

We can use the quotient of powers, \dfrac{{a}^\frac{m}{n}}{{a}^\frac{p}{n}}=a^\frac{m-p}{n}, to simplify the expression.

Solution

\displaystyle \dfrac{4^{- \frac{2}{9}}}{4^{- \frac{2}{9}}}\displaystyle =\displaystyle 4^\frac{-2-(-2)}{9}Quotient of powers law
\displaystyle =\displaystyle 4^\frac{0}{9}Evaluate the subtraction
\displaystyle =\displaystyle 4^0Evaluate the division
\displaystyle =\displaystyle 1Definition of zero exponent

Reflection

It is possible to simplify this expression more quickly by recognizing that a number divided by itself always results in 1.

c

\left(5^{0}\right)^{-2} \cdot \left(5^{ - \frac{7}{17} }\right)^{0}

Approach

We can use the definition of zero exponents to simplify the expression, then apply the law of negative exponents followed by multiplication to fully evaluate the expression.

Solution

\displaystyle \left(5^{0}\right)^{-2} \cdot \left(5^{ - \frac{7}{17} }\right)^{0}\displaystyle =\displaystyle 1^{-2} \cdot 1Definiton of zero exponent
\displaystyle =\displaystyle \frac{1}{1^2} \cdot 1Definition of negative exponent
\displaystyle =\displaystyle \frac{1}{1} \cdot 1Evaluate the exponent
\displaystyle =\displaystyle 1 \cdot 1Evaluate the division
\displaystyle =\displaystyle 1Evaluate the multiplication

Outcomes

MA.912.NSO.1.1

Extend previous understanding of the Laws of Exponents to include rational exponents. Apply the Laws of Exponents to evaluate numerical expressions and generate equivalent numerical expressions involving rational exponents.

MA.912.NSO.1.2

Generate equivalent algebraic expressions using the properties of exponents.

MA.912.NSO.1.4

Apply previous understanding of operations with rational numbers to add, subtract, multiply and divide numerical radicals.

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