Rational Exponents and Radicals

Rational exponents and radicals are two different ways to express the same quantities. Sometimes, one form is more useful than the other. Thus, being able to translate between the two is important.

Rule

Rational Exponents

When a number is raised to the power of a fraction, that fraction is the number's rational exponent. Such an expression is equivalent to a root.

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

Notice that the denominator of the rational exponent gives the root, while the numerator gives the power to which

a

is raised. The Properties of Exponents apply to rational exponents in the same way they apply to integers. Consider

\sqrt[5]{8^2}.

\sqrt[5]{8^2}

RootToPowD

\sqrt[n]{a}=a^{\frac{1}{n}}

\left(8^2\right)^{\frac 1 5}

PowPow

\left(a^{m}\right)^{n}=a^{m\cdot n}

8^{2\cdot \frac 1 5}

MoveLeftFacToNumOne

a\cdot \dfrac{1}{b}= \dfrac{a}{b}

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

Thus,

\sqrt[5]{8^2}

is equivalent to

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

Rewrite between rational exponent form and radical form

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Exercise

Rewrite the given expressions in the opposite form. $\sqrt[5]{x} \quad \text{and} \quad x^{\frac{2}{3}}$

Show Solution

Solution

Example

$\sqrt[5]{x}$

To begin, notice that the expression $\sqrt[5]{x}$ is written in radical form. Thus, it needs to be rewritten with a rational exponent. Recall that $\sqrt[n]{a^m}=a^{\frac{m}{n}}.$ Notice that in $\sqrt[5]{x},$ $x$ does not have an exponent. That means it is raised to the power of $1.$ This gives $\sqrt[5]{x}=x^{\frac{1}{5}}.$

Example

$x^{\frac{2}{3}}$

This expression has a rational exponent. Thus, we must rewrite it with a radical. Recall that $a^{\frac{m}{n}}=\sqrt[n]{a^m}.$ Since the denominator of the exponent is $3,$ we can write the cube root of $x^2.$ $x^{\frac{2}{3}}=\sqrt[3]{x^2}$

Rule

Properties of Radicals

The properties of radicals allow expressions with radicals to be rewritten.

Rule

Product Property of Radicals

Rule

\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot \sqrt[n]{b}

The

n

th root of a product can be written as the product of the

n

th root of each factor. For example,

\sqrt[3]{7\cdot 9},

can be expressed as follows using this rule.

\sqrt[3]{7\cdot 9}=\sqrt[3]{7}\cdot \sqrt[3]{9}

This rule can be explained by expressing the radical using a rational exponent, and then using the Power of a Product Property.

\sqrt[3]{7\cdot 9}

RootToPowSL

\sqrt[n]{a}=a^{1/n}

(7\cdot 9)^{1/3}

PowProd

\left(a \cdot b\right)^{m}=a^m\cdot b^m

7^{1/3}\cdot 9^{1/3}

PowToRootSL

a^{1/n}=\sqrt[n]{a}

\sqrt[3]{7}\cdot \sqrt[3]{9}

This rule is valid for all values of

a,

b

and

n.

However, in order to avoid non-real solutions it is necessary that

a\geq 0

and

b\geq 0

for even values of

n.

Rule

Quotient Property of Radicals

Rule

\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}

The

n

th root of a fraction can be written as the

n

th root of the numerator divided by the

n

th root of the denominator. For example,

\sqrt[3]{\frac{7}{9}},

can be expressed as follows using this rule.

\sqrt[3]{\dfrac{7}{9}}=\dfrac{\sqrt[3]{7}}{\sqrt[3]{9}}

This rule can be explained by rewriting the radical as a rational exponent, and then using the Power of a Quotient Property.

\sqrt[3]{\dfrac{7}{9}}

RootToPowSL

\sqrt[n]{a}=a^{1/n}

\left( \dfrac{7}{9} \right)^{1/3}

PowQuot

\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}

\dfrac{7^{1/3}}{9^{1/3}}

PowToRootSL

a^{1/n}=\sqrt[n]{a}

\dfrac{\sqrt[3]{7}}{\sqrt[3]{9}}

This rule is valid for all values of

a

and

n,

and when

b\neq 0.

However, in order to avoid non-real solutions it is necessary that

a\geq 0

and

b> 0

for even values of

n.

Simplify the expression using the properties of radicals

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Exercise

Simplify the expression using the properties of radicals. $\dfrac{\sqrt{3}\cdot \sqrt{15}}{\sqrt{5}}$

Show Solution

Solution

When two radicals with the same index are either multiplied or divided they can be simplified using the properties of radicals. We'll begin by simplifying the numerator using the Product Property of Radicals.

\dfrac{\sqrt{3}\cdot \sqrt{15}}{\sqrt{5}}

ProdSqrt

\sqrt{a}\cdot\sqrt{b}=\sqrt{a\cdot b}

\dfrac{\sqrt{3\cdot 15}}{\sqrt{5}}

MultiplyMultiply

\dfrac{\sqrt{45}}{\sqrt{5}}

Next we simplify the expression using the Quotient Property of Radicals.

\dfrac{\sqrt{45}}{\sqrt{5}}

DivSqrt

\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}

\sqrt{\dfrac{45}{5}}

CalcQuotCalculate quotient

\sqrt{9}

CalcRootCalculate root

3

Thus, the expression simplifies to

3.

Rule

Simplifying an $n^{\text{th}}$ Root

In order to simplify an $n^{\text{th}}$ root, it is necessary that the radicand can be expressed as a power. If the index of the radical and the power of the radicand are equal, $\sqrt[{\color{#0000FF}{n}}]{a^{\color{#0000FF}{n}}},$ the radical expression can be simplified as follows. $\sqrt[n]{a^n}= \begin{cases} \phantom{|}a\phantom{|} \text{ if } n \text{ is odd}\\ |a| \text{ if } n \text{ is even} \end{cases}$

The absolute value of a number is always non-negative, so when

n

is even, the result will always be non-negative.

Method

Simplifying Radical Expressions

Expressions with radicals can be written using rational exponents. Then, they can be simplified using the properties of exponents. Consider the following expression.

\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^4}}{\sqrt[6]{x^7}}

1

Rewrite terms into

\sqrt[n]{a^n}

When a term inside a radical has a power greater than the index of the radical, it can be rewritten into

\sqrt[n]{a^n}

-form. In the example, there are two such terms,

\sqrt[3]{x^4}

and

\sqrt[6]{x^7}.

4>3 \quad \text{and} \quad 7>6

First, the Product of Powers Property can be used to rewrite the terms under these radicals.

\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^4}}{\sqrt[6]{x^7}}

RewriteRewrite

4

as

{\color{#FF0000}{3+1}}

\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^{{\color{#FF0000}{3+1}}}}}{\sqrt[6]{x^7}}

RewriteRewrite

7

as

{\color{#0000FF}{6+1}}

\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^{3+1}}}{\sqrt[6]{x^{{\color{#0000FF}{6+1}}}}}

SumInExponent

a^{m+n}=a^{m}\cdot{a}^{n}

\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3\cdot x^1}}{\sqrt[6]{x^6\cdot x^1}}

The products under the radicals can now be rewritten using the Product Property of Radicals.

\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3\cdot x^1}}{\sqrt[6]{x^6\cdot x^1}}

RootProd

\sqrt[3]{a\cdot b}=\sqrt[3]{a}\cdot\sqrt[3]{b}

\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3}\cdot \sqrt[3]{x^1}}{\sqrt[6]{x^6\cdot x^1}}

RootProd

\sqrt[6]{a\cdot b}=\sqrt[6]{a}\cdot\sqrt[6]{b}

\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3}\cdot \sqrt[3]{x^1}}{\sqrt[6]{x^6}\cdot \sqrt[6]{x^1}}

ExponentOne

a^1=a

\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3}\cdot \sqrt[3]{x}}{\sqrt[6]{x^6}\cdot \sqrt[6]{x}}

2

Simplify

\sqrt[n]{a^n}

-terms

The

\sqrt[n]{a^n}

-terms can now be simplified as follows.

\sqrt[n]{a^n}= \begin{cases} \phantom{|}a\phantom{|} \text{ if } n \text{ is odd}\\ |a| \text{ if } n \text{ is even} \end{cases}

In the expression there are two terms that can be simplified using this rule.

\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3}\cdot \sqrt[3]{x}}{\sqrt[6]{x^6}\cdot \sqrt[6]{x}}

RootPowToNumber

\sqrt[3]{a^3}=a

\dfrac{\sqrt[4]{x^3}\cdot x\cdot \sqrt[3]{x}}{\sqrt[6]{x^6}\cdot \sqrt[6]{x}}

\sqrt[6]{a^6}=|a|

\dfrac{\sqrt[4]{x^3}\cdot x\cdot \sqrt[3]{x}}{|x|\cdot \sqrt[6]{x}}

3

Express the radicals using rational exponents

Next, rewrite the radicals using rational exponents. The example can be rewritten as follows. $\dfrac{\sqrt[4]{x^3}\cdot x\cdot \sqrt[3]{x}}{|x| \cdot \sqrt[6]{x}} \quad \Leftrightarrow \quad \dfrac{x^{\frac{3}{4}}\cdot x\cdot x^{\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}$

4

Simplify using the laws of exponents

When the expression is written using rational exponents, it can be simplified. When two terms with the same base are multiplied, the exponents are added according to the Product of Powers Property.

\dfrac{x^{\frac{3}{4}}\cdot x \cdot x^{\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}

a=a^1

\dfrac{x^{\frac{3}{4}}\cdot x^1 \cdot x^{\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}

MultPow

a^{m}\cdot{a}^{n}=a^{m+n}

\dfrac{x^{\frac{3}{4}+1+\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}

To simplify the exponent further, which requires adding and subtracting fractions, the denominators must be made the same. Here, the least common denominator is

12.

\dfrac{x^{\frac{3}{4}+1+\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}

ExpandFrac

\dfrac{a}{b}=\dfrac{a \cdot 3}{b \cdot 3}

\dfrac{x^{\frac{9}{12}+1+\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}

OneToFracRewrite

1

as

\dfrac{12}{12}

\dfrac{x^{\frac{9}{12}+\frac{12}{12}+\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}

ExpandFrac

\dfrac{a}{b}=\dfrac{a \cdot 4}{b \cdot 4}

\dfrac{x^{\frac{9}{12}+\frac{12}{12}+\frac{4}{12}}}{|x| \cdot x^{\frac{1}{6}}}

AddFracAdd fractions

\dfrac{x^{\frac{25}{12}}}{|x| \cdot x^{\frac{1}{6}}}

Since the expressions in the numerator and the denominator have the same bases,

x,

they can be simplified. First, by using the Quotient of Powers Property, the expression is written as one term. To simplify the exponent, the denominators must then be made the same.

\dfrac{x^{\frac{25}{12}}}{|x| \cdot x^{\frac{1}{6}}}

DivPow

\dfrac{a^m}{a^n}= a^{m-n}

\dfrac{x^{\frac{25}{12}-\frac{1}{6}}}{|x|}

ExpandFrac

\dfrac{a}{b}=\dfrac{a \cdot 2}{b \cdot 2}

\dfrac{x^{\frac{13}{12}-\frac{2}{12}}}{|x|}

SubFracSubtract fractions

\dfrac{x^{\frac{11}{12}}}{|x|}

5

Rewrite expression using radicals

When the expression has been simplified completely, it can be rewritten using radicals. $\dfrac{x^{\frac{11}{12}}}{|x|}=\dfrac{\sqrt[12]{x^{11}}}{|x|}$ This is the simplified expression.

Simplify the expression with rational exponents

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Exercise

Simplify the expression using the properties of exponents. $\dfrac{2^{\frac{10}{3}}}{4^{\frac{4}{3}}\cdot 16^{\frac{1}{3}}}$

Show Solution

Solution

Notice that the terms in this expression each have a difference base.

2, \quad 4, \quad \text{and} \quad 16.

This expression cannot be simplified until the bases are the same. Therefore, we can begin by rewriting the terms with base

2.

Notice that

4=2^2

and

16=2^4.

\dfrac{2^{\frac{10}{3}}}{4^{\frac{4}{3}}\cdot 16^{\frac{1}{3}}}

RewriteRewrite

4

as

2^2

\dfrac{2^{\frac{10}{3}}}{\left(2^2 \right)^{\frac{4}{3}}\cdot 16^{\frac{1}{3}}}

RewriteRewrite

16

as

2^4

\dfrac{2^{\frac{10}{3}}}{\left(2^2 \right)^{\frac{4}{3}}\cdot \left(2^4 \right)^{\frac{1}{3}}}

Next, we can rewrite the expression using the Power of a Power Property.

\dfrac{2^{\frac{10}{3}}}{\left(2^2 \right)^{\frac{4}{3}}\cdot \left(2^4 \right)^{\frac{1}{3}}}

PowPow

\left(a^{m}\right)^{n}=a^{m\cdot n}

\dfrac{2^{\frac{10}{3}}}{2^{2 \cdot \frac{4}{3}}\cdot 2^{4 \cdot \frac{1}{3}}}

MultiplyMultiply

\dfrac{2^{\frac{10}{3}}}{2^{\frac{8}{3}}\cdot 2^{\frac{4}{3}}}

All terms in the expression now have the same base. From here, we can use the Product of Powers Property and the Quotient of Powers Property to simplify.

\dfrac{2^{\frac{10}{3}}}{2^{\frac{8}{3}}\cdot 2^{\frac{4}{3}}}

MultPow

a^{m}\cdot{a}^{n}=a^{m+n}

\dfrac{2^{\frac{10}{3}}}{2^{\frac{8}{3}+\frac{4}{3}}}

AddFracAdd fractions

\dfrac{2^{\frac{10}{3}}}{2^{\frac{12}{3}}}

DivPow

\dfrac{a^m}{a^n}= a^{m-n}

2^{\frac{10}{3}-\frac{12}{3}}

SubFracSubtract fractions

2^{\text{-} \frac{2}{3}}

The answer is

2^{\text{-} \frac{2}{3}},

which can also be written as

\dfrac{1}{2^{\frac{2}{3}}}.