Previously, we have seen that we can rewrite square roots and cube roots with rational exponents. Since \left(a^\frac{1}{2}\right)^2 =a and \left(\sqrt{a}\right)^2=a, that means a^\frac{1}{2}=\sqrt{a}. A similar argument can be used for cube roots.

Image showing how the exponential form base a raised to the rational exponent one-third is equal to the radical form cube root of a where 3 is the index and a is the radicand inside the radical sign

Exploration

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

Fill in the blanks: 81^\frac{1}{⬚}=3=\sqrt[⬚]{81}
What values for the missing exponents would make each statement true?
What is the relationship between the values from each statement?
Explain the meaning of 81^\frac{3}{4}.

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, using rational exponents 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:

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

\bm{a}

is the base or radicand

\bm{m}

is the numerator or power

\bm{n}

is the denominator or index

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

\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}}

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 expression in an equivalent form using rational exponents. Assume all variables are positive.

\sqrt[5]{x^7}

Worked Solution

Create a strategy

We can write this expression using rational exponents 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.

The variable x is raised to a fractional exponent m over n. tha numerator m is labeled power and the denominator n is labeled index

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

Worked Solution

Create a strategy

After rewriting the radical with rational exponents, we can use a few exponent properties to simplify the expression. First, we can use the power of a product property to apply the rational exponent to both bases. We can then use the power of a power property 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. Had it not mentioned that the variables were positive, we would have needed to consider complex values when simplifying.

Example 3

Write the following expressions in simplified radical form.

\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=-4 and c=24, this expression becomes: \sqrt[5]{-4\cdot 24}=\sqrt[5]{-96}We can use the fact that \left(-1\right)^5=-1 to rewrite the radical.

\displaystyle \sqrt[5]{-96}	\displaystyle =	\displaystyle \sqrt[5]{-1\cdot 96}
	\displaystyle =	\displaystyle \sqrt[5]{\left(-1\right)^5\cdot 96}
	\displaystyle =	\displaystyle -\sqrt[5]{96}

Next, we can use prime factorization to simplify this further:

\displaystyle -\sqrt[5]{96}	\displaystyle =	\displaystyle -\sqrt[5]{2\cdot2\cdot2\cdot2\cdot2\cdot3}	Find prime factorization of 96
\displaystyle \text{ }	\displaystyle =	\displaystyle -\sqrt[5]{2^5\cdot3}	Rewrite using exponents
\displaystyle \text{ }	\displaystyle =	\displaystyle -\sqrt[5]{2^5}\cdot\sqrt[5]{3}	Product of radicals property
\displaystyle \text{ }	\displaystyle =	\displaystyle -2\sqrt[5]{3}	Evalutate the fifth root

When the index of a radical is odd, the variables do not need to be limited to positive values.

\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}}\left(y^8\right)^{\frac{1}{4}}}	Power of a product
	\displaystyle =	\displaystyle \frac{\sqrt[4]{81}\sqrt[4]{x}}{\sqrt[4]{625}\sqrt[4]{y^8}}	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

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

Worked Solution

Create a strategy

To simplify, we will apply the power of a product property. After, we can use the power of a power property to simplify further.

Apply the idea

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

(-12x^5y^3)^{-\frac{1}{2}} assume all variables are positive.

Worked Solution

Create a strategy

To simplify, we will apply the negative exponent property, rewrite the expression in radical form, then simplify the radical if possible

Apply the idea

\displaystyle (-12x^5y^3)^{-\frac{1}{2}}	\displaystyle =	\displaystyle \dfrac{1}{(-12x^5y^3)^{\frac{1}{2}}}	Negative exponent property
	\displaystyle =	\displaystyle \dfrac{1}{\left(-12\right)^{\frac{1}{2}}x^{\frac{5}{2}}y^{\frac{3}{2}}}	Power of a power rule
	\displaystyle =	\displaystyle \dfrac{1}{\sqrt{-12}\cdot \sqrt{x^5}\cdot \sqrt{y^3}}	Rewrite in radical form
	\displaystyle =	\displaystyle \dfrac{1}{2i\sqrt{3}\cdot x^2\sqrt{x}\cdot y\sqrt{y}}	Simplify each radical
	\displaystyle =	\displaystyle \dfrac{1}{2x^2yi\sqrt{3xy}}	Rewrite using a single radical

Idea summary

We can write radicals using rational exponents, for integer values of m and n, where n\neq 0:

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

\bm{a}

is the base or radicand

\bm{m}

is the numerator or power

\bm{n}

is the denominator or index

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

A2.EO.2

The student will perform operations on and simplify radical expressions.

A2.EO.2c

Convert between radical expressions and expressions containing rational exponents.

4.02 Rational exponents

Ideas

Rational exponents

Exploration

Examples

Example 1

Example 2

Example 3

Outcomes

A2.EO.2

A2.EO.2c

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