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1. Reasoning with Quantities and Expressions
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United States of America
Grades 9-12

1.04 Rational exponents

Lesson
Practice

Introduction

Properties of exponents and solving equations with square roots and cube roots were topics covered in 8th grade. This lesson will combine the two by defining a radical as a rational exponent and applying the properties of exponents to simplify expressions.

Rational exponents

Remember that when multiplying a number by itself repeatedly, we are able to use exponent notation to write the expression more simply.

For example, a multiplied by itself m times can be written in exponential form as:

\displaystyle a^m
\bm{a}
is the base.
\bm{m}
is the exponent.

Recall the following properties of integer exponents that developed from this:

\text{Product of powers}a^{m} \cdot a^{n} = a^{m+n}
\text{Quotient of powers}\dfrac{{a}^{m}}{{a}^{n}}=a^{m-n}
\text{Power of a power}\left(a^{m}\right)^{n} = a^{mn}
\text{Power of a product}\left(a b\right)^{m} = a^{m} \cdot b^{m}
\text{Power of a quotient}\left(\dfrac{a} {b}\right)^{m} =\dfrac {a^{m}} {b^{m}}
\text{Identity exponent}a^1=a
\text{Zero exponent}a^0=1
\text{Negative exponent}a^{- {m}}=\dfrac{1}{a^{m}}

Exploration

Consider the following statements:\begin{aligned} 3 \times 3 \times 3 \times 3 &= 3^{\text{⬚}}=81\\ 81^{\text{⬚}} &= 3\\ 81^\frac{1}{4} \times 81^\frac{1}{4} \times 81^\frac{1}{4} \times 81^\frac{1}{4} &= 81^{\text{⬚}}=3^{\text{⬚}}\\ \left(\sqrt[4]{81}\right)^4&=\left(81^{\text{⬚}}\right)^4=81\\ \left(\sqrt[4]{81}\right)^3\times 81^\frac{1}{4}&=81^{\text{⬚}} \times 81^\frac{1}{4}=81 \end{aligned}

  1. Which exponent property is being applied?
  2. What values for the missing exponents would make each statement true?
  3. What is the relationship between the values from each statement?
  4. Explain the meaning of 81^\frac{3}{4}.

The same properties of exponents that we use for integer exponents can also be applied to exponents that are fractions. We call these rational exponents.

Using the properties of exponents, we can express a^\frac{1}{n}, which represents one of n equal factors whose product equals a, multiplied by itself m times, in exponential form as:

\displaystyle a^\frac{m}{n}
\bm{a}
is the base.
\bm{\frac{m}{n}}
is the exponent.

where, m and n are integers, and n \neq 0.

In general, a radical can also be rewritten as a rational exponent in the following ways:

\left(\sqrt[n]{a}\right)^m=a^\frac{m}{n}

\sqrt[n]{a^m}=a^\frac{m}{n}

Notice that the exponent, m, becomes the numerator of the rational exponent, and the index of the radical, n, becomes the denominator of the rational exponent.

We can use these rules for rewriting radicals along with the properties of exponents to simplify expressions involving radicals and rational exponents.

Examples

Example 1

Use the properties of exponents to define a rational exponent that would make the statement true: \left(\sqrt[3]{x^7}\right)^3=\left(x^{\frac{⬚}{⬚}}\right)^3=x^7

Worked Solution
Create a strategy

To get from the second step to the last step, the power of a power property is being used. This property tells us to multiply the exponent inside the parenthesis with the exponent outside the parenthesis. We need a fraction that when multiplied by 3 will result in 7.

Apply the idea

\left(\sqrt[3]{x^7}\right)^3=\left(x^{\frac{7}{3}}\right)^3=x^7

Reflect and check

Checking the multiplication in the exponents:

\dfrac{7}{3}\cdot 3=7

This shows the radical in the first expression can be rewritten in rational exponent form as shown in the second expression: \sqrt[3]{x^7}=x^{\frac{7}{3}}

Example 2

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

a

\sqrt[5]{x^7}

Worked Solution
Create a strategy

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

Apply the idea

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

Reflect and check

This image can help us visualize the rule in another way. The exponent became the numerator of the rational exponent, and the index became the denominator.

A figure showing the expression x raised to m divided by n. The variable m is labeled as power, variable n is labeled as root.
b

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

Worked Solution
Create a strategy

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 rational exponents. We can then use the power of a power law to simplify the powers.

Apply the idea
\displaystyle \left(\sqrt[4]{x^3y^5}\right)^{12} \displaystyle =\displaystyle \left(x^{\frac{3}{4}}y^{\frac{5}{4}}\right)^{12}Rewrite using rational exponents
\displaystyle =\displaystyle \left(x^\frac{3}{4}\right)^{12}\left(y^\frac{5}{4}\right)^{12}Power of a product
\displaystyle =\displaystyle x^{\left(\frac{3}{4}\cdot12\right)}y^{\left(\frac{5}{4}\cdot12\right)}Power of a power
\displaystyle =\displaystyle x^9y^{15}Simplify the exponents
Reflect and check

In the instructions, it said "assume all variables are positive." This is important for this problem because the 4 \text{th} root of a negative number is undefined. Because negative numbers would be undefined, we would not be able to apply these properties and simplify the expression.

Example 3

Write the following expressions in reduced radical form.

a

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

Worked Solution
Create a strategy

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

Apply the idea

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

Reflect and check

The instructions for this problem did not say that the variables needed to represent positive numbers. That is because the odd root of a negative number is defined. For example, if b=-2 and c=16, this expression becomes:

\sqrt[5]{-32}=-2

This is a real number, so the variables do not need to represent only positive numbers.

b

\left(\dfrac{81x}{625y^8}\right)^{\frac{1}{4}} assume all variables are positive.

Worked Solution
Create a strategy

We are going to use the properties of exponents to simplify this expression. There is nothing that can be simplified within the parenthesis, so we can begin by applying the power to both the numerator and denominator.

Apply the idea
\displaystyle \left(\dfrac{81x}{625y^8}\right)^{\frac{1}{4}}\displaystyle =\displaystyle \dfrac{\left(81x\right)^{\frac{1}{4}}}{\left(625y^8\right)^{\frac{1}{4}}}Power of a quotient
\displaystyle =\displaystyle \dfrac{81^{\frac{1}{4}} x^{\frac{1}{4}}}{625^{\frac{1}{4}}y^2}Power of a power
\displaystyle =\displaystyle \frac{\sqrt[4]{81}\sqrt[4]{x}}{\sqrt[4]{625}y^2}Rewrite in radical form
\displaystyle =\displaystyle \frac{3\sqrt[4]{x}}{5y^2}Evaluate the radicals
Reflect and check

Using radicals to simplify this expression would result in the same answer.

\displaystyle \left(\dfrac{81x}{625y^8}\right)^\frac{1}{4}\displaystyle =\displaystyle \sqrt[4]{\dfrac{81x}{625y^8}}Rewrite in radical form
\displaystyle =\displaystyle \dfrac{\sqrt[4]{81x}}{\sqrt[4]{625y^8}}Quotient of radicals
\displaystyle =\displaystyle \dfrac{\sqrt[4]{81}\sqrt[4]{x}}{\sqrt[4]{625}\sqrt[4]{y^8}}Product of radicals
\displaystyle =\displaystyle \dfrac{3\sqrt[4]{x}}{5y^2}Evaluate the radicals

Example 4

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}}

Worked Solution
Create a strategy

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.

Apply the idea
\displaystyle 2c^{\frac{1}{5}}\cdot3c^{\frac{2}{5}}\displaystyle =\displaystyle 2\left(3\right)c^{\frac{1}{5}}c^{\frac{2}{5}}Multiply the 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
Reflect and check

If we wanted to write the answer in radical form, it would be 6\sqrt[5]{c^3}.

b

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

Worked Solution
Create a strategy

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.

Apply the idea
\displaystyle \dfrac{15 p^{\frac{7}{3}}}{5 p^{\frac{1}{6}}}\displaystyle =\displaystyle 3\dfrac{ p^{\frac{7}{3}}}{ p^{\frac{1}{6}}}Divide the coefficients
\displaystyle =\displaystyle 3p^{\frac{7}{3}-\frac{1}{6}}Quotient of powers
\displaystyle =\displaystyle 3p^{\frac{14}{6}-\frac{1}{6}}Find a common denominator
\displaystyle =\displaystyle 3p^{\frac{13}{6}}Simplify the exponent
Reflect and check

Whenever the rational exponent is an improper fraction, it would be reduced when we switch to radical form.

3p^{\frac{13}{6}}=3p^2\sqrt[6]{p}

c

\left(-27a^{15}b^{27}\right)^\frac{1}{3}

Worked Solution
Create a strategy

To simplify, we will apply the power of products property.

Apply the idea
\displaystyle \left(-27a^{15}b^{27}\right)^\frac{1}{3}\displaystyle =\displaystyle \left(-27\right)^\frac{1}{3}a^5b^9Power of a product
\displaystyle =\displaystyle \sqrt[3]{-27}a^5b^9Rewrite in radical form
\displaystyle =\displaystyle -3a^5b^9Evaluate the radical

Example 5

Fully simplify the following expressions. Write the result in reduced radical form. Assume all variables are positive.

a

\left(2a^{\frac{1}{4}}\right)\left(16a\right)^{\frac{1}{2}}

Worked Solution
Create a strategy

We can first evaluate the rational exponent 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.

Apply the idea
\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
Reflect and check

In the second to last step, we had 8a^{\frac{3}{4}}. Notice that the exponent is only applied to the base of a, not the coefficient. That is why 8 does not go underneath the radical.

b

\left(\dfrac{m^{\frac{5}{2}}n^{-\frac{1}{4}}}{m^{-1}n}\right)^{0.8}

Worked Solution
Create a strategy

The rational exponents are written as fractions and decimals, but it is easiest to work with them when they are in the same form. We can begin by rewriting 0.8 as \dfrac{4}{5}.

Apply the idea
\displaystyle \left(\dfrac{m^{\frac{5}{2}}n^{-\frac{1}{4}}}{m^{-1}n}\right)^{0.8}\displaystyle =\displaystyle \left(\dfrac{m^{\frac{5}{2}}n^{-\frac{1}{4}}}{m^{-1}n}\right)^{\frac{4}{5}}Rewrite rational exponent as a decimal
\displaystyle =\displaystyle \left(m^{\frac{5}{2}-(-1)}n^{-\frac{1}{4}-1}\right)^{\frac{4}{5}}Quotient of powers
\displaystyle =\displaystyle \left(m^{\frac{7}{2}}n^{-\frac{5}{4}}\right)^{\frac{4}{5}}Evaluate the subtraction in the exponents
\displaystyle =\displaystyle m^{\frac{14}{5}}n^{-1}Power of a product
\displaystyle =\displaystyle \dfrac{m^{\frac{14}{5}}}{n}Negative exponent property
Reflect and check

We could have applied the properties in a different order, and it would have resulted in the same answer.

\displaystyle \left(\dfrac{m^{\frac{5}{2}}n^{-\frac{1}{4}}}{m^{-1}n}\right)^{0.8}\displaystyle =\displaystyle \left(\dfrac{m^{\frac{5}{2}}n^{-\frac{1}{4}}}{m^{-1}n}\right)^{\frac{4}{5}}Rewrite rational exponent as a decimal
\displaystyle =\displaystyle \dfrac{m^{\frac{5}{2}\cdot \frac{4}{5}}n^{-\frac{1}{4}\cdot \frac{4}{5}}}{m^{-1\cdot \frac{4}{5}}n^{\frac{4}{5}}}Power of powers
\displaystyle =\displaystyle \dfrac{m^{2}n^{-\frac{1}{5}}}{m^{-\frac{4}{5}}n^{\frac{4}{5}}}Evaluate the powers
\displaystyle =\displaystyle m^{2-\left(-\frac{4}{5}\right)}n^{-\frac{1}{5}-\frac{4}{5}}Quotient of powers
\displaystyle =\displaystyle m^{\frac{14}{5}}n^{-1}Evaluate the exponents
\displaystyle =\displaystyle \dfrac{m^\frac{14}{5}}{n}Negative exponent property
c

\sqrt{50x^6y^5}

Worked Solution
Create a strategy

We can use rational exponents to simplify this expression.

Apply the idea
\displaystyle \sqrt{50x^6y^5}\displaystyle =\displaystyle \left(50x^6y^5\right)^{\frac{1}{2}}Rewrite in exponential form
\displaystyle =\displaystyle 50^{\frac{1}{2}} \left(x^6\right)^{\frac{1}{2}} \left(y^5\right)^{\frac{1}{2}}Power of a product
\displaystyle =\displaystyle \left(25\cdot 2\right)^{\frac{1}{2}} \left(x^6\right)^{\frac{1}{2}} \left(y^4\cdot y\right)^{\frac{1}{2}}Separate into factors
\displaystyle =\displaystyle 25^{\frac{1}{2}}\cdot 2^{\frac{1}{2}}\cdot \left(x^6\right)^{\frac{1}{2}} \left(y^4\right)^{\frac{1}{2}} y^{\frac{1}{2}}Power of a product
\displaystyle =\displaystyle 25^{\frac{1}{2}}\cdot 2^{\frac{1}{2}} x^3 y^2 y^{\frac{1}{2}}Power of a power
\displaystyle =\displaystyle 25^{\frac{1}{2}} x^3 y^2\cdot 2^{\frac{1}{2}}y^{\frac{1}{2}}Commutative property
\displaystyle =\displaystyle \sqrt{25} x^3 y^2\cdot 2^{\frac{1}{2}}y^{\frac{1}{2}}Rewrite in radical form
\displaystyle =\displaystyle 5x^3y^2\cdot \left(2y\right)^{\frac{1}{2}}Power of a power
\displaystyle =\displaystyle 5x^3y^2\sqrt{2y}Rewrite in radical form
Idea summary

The properties of integer exponents can also be applied to rational exponents:

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

Outcomes

N.RN.A.1

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

N.RN.A.2

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

A.SSE.B.3.C

Use the properties of exponents to transform expressions for exponential functions.

F.IF.C.8.B

Use the properties of exponents to interpret expressions for exponential functions.

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