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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.

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.


Notice that the denominator of the rational exponent gives the root, while the numerator gives the power to which aa is raised. The Properties of Exponents apply to rational exponents in the same way they apply to integers. Consider 825.\sqrt[5]{8^2}.
(82)15\left(8^2\right)^{\frac 1 5}
82158^{2\cdot \frac 1 5}
Thus, 825\sqrt[5]{8^2} is equivalent to 825.8^{\frac {2}{5}}.

Rewrite the given expressions in the opposite form. x5andx23 \sqrt[5]{x} \quad \text{and} \quad x^{\frac{2}{3}}

Show Solution


To begin, notice that the expression x5\sqrt[5]{x} is written in radical form. Thus, it needs to be rewritten with a rational exponent. Recall that amn=amn.\sqrt[n]{a^m}=a^{\frac{m}{n}}. Notice that in x5,\sqrt[5]{x}, xx does not have an exponent. That means it is raised to the power of 1.1. This gives x5=x15. \sqrt[5]{x}=x^{\frac{1}{5}}.



This expression has a rational exponent. Thus, we must rewrite it with a radical. Recall that amn=amn.a^{\frac{m}{n}}=\sqrt[n]{a^m}. Since the denominator of the exponent is 3,3, we can write the cube root of x2.x^2. x23=x23 x^{\frac{2}{3}}=\sqrt[3]{x^2}


Properties of Radicals

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


Product Property of Radicals


abn=anbn\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot \sqrt[n]{b}
The nnth root of a product can be written as the product of the nnth root of each factor. For example, 793,\sqrt[3]{7\cdot 9}, can be expressed as follows using this rule. 793=7393 \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.
793\sqrt[3]{7\cdot 9}
(79)1/3(7\cdot 9)^{1/3}
71/391/37^{1/3}\cdot 9^{1/3}
7393\sqrt[3]{7}\cdot \sqrt[3]{9}
This rule is valid for all values of a,a, bb and n.n. However, in order to avoid non-real solutions it is necessary that a0a\geq 0 and b0b\geq 0 for even values of n.n.

Quotient Property of Radicals


The nnth root of a fraction can be written as the nnth root of the numerator divided by the nnth root of the denominator. For example, 793,\sqrt[3]{\frac{7}{9}}, can be expressed as follows using this rule. 793=7393 \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.
(79)1/3\left( \dfrac{7}{9} \right)^{1/3}
This rule is valid for all values of aa and n,n, and when b0.b\neq 0. However, in order to avoid non-real solutions it is necessary that a0a\geq 0 and b>0b> 0 for even values of n.n.

Simplify the expression using the properties of radicals. 3155 \dfrac{\sqrt{3}\cdot \sqrt{15}}{\sqrt{5}}

Show 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.
3155\dfrac{\sqrt{3}\cdot \sqrt{15}}{\sqrt{5}}
3155\dfrac{\sqrt{3\cdot 15}}{\sqrt{5}}
Next we simplify the expression using the Quotient Property of Radicals.
Thus, the expression simplifies to 3.3.

Simplifying an nthn^{\text{th}} Root

In order to simplify an nthn^{\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, ann,\sqrt[{\color{#0000FF}{n}}]{a^{\color{#0000FF}{n}}}, the radical expression can be simplified as follows. ann={a if n is odda if n is even \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 nn is even, the result will always be non-negative.

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. x34x43x76 \dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^4}}{\sqrt[6]{x^7}}


Rewrite terms into ann\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 ann\sqrt[n]{a^n}-form. In the example, there are two such terms, x43\sqrt[3]{x^4} and x76.\sqrt[6]{x^7}. 4>3and7>6 4>3 \quad \text{and} \quad 7>6 First, the Product of Powers Property can be used to rewrite the terms under these radicals.
x34x43x76\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^4}}{\sqrt[6]{x^7}}
x34x3+13x76\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^{{\color{#FF0000}{3+1}}}}}{\sqrt[6]{x^7}}
x34x3+13x6+16\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^{3+1}}}{\sqrt[6]{x^{{\color{#0000FF}{6+1}}}}}
x34x3x13x6x16\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.
x34x3x13x6x16\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3\cdot x^1}}{\sqrt[6]{x^6\cdot x^1}}
x34x33x13x6x16\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3}\cdot \sqrt[3]{x^1}}{\sqrt[6]{x^6\cdot x^1}}
x34x33x13x66x16\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3}\cdot \sqrt[3]{x^1}}{\sqrt[6]{x^6}\cdot \sqrt[6]{x^1}}
x34x33x3x66x6\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3}\cdot \sqrt[3]{x}}{\sqrt[6]{x^6}\cdot \sqrt[6]{x}}


Simplify ann\sqrt[n]{a^n}-terms
The ann\sqrt[n]{a^n}-terms can now be simplified as follows. ann={a if n is odda if n is even \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.
x34x33x3x66x6\dfrac{\sqrt[4]{x^3}\cdot \sqrt[3]{x^3}\cdot \sqrt[3]{x}}{\sqrt[6]{x^6}\cdot \sqrt[6]{x}}
x34xx3x66x6\dfrac{\sqrt[4]{x^3}\cdot x\cdot \sqrt[3]{x}}{\sqrt[6]{x^6}\cdot \sqrt[6]{x}}
x34xx3xx6\dfrac{\sqrt[4]{x^3}\cdot x\cdot \sqrt[3]{x}}{|x|\cdot \sqrt[6]{x}}


Express the radicals using rational exponents

Next, rewrite the radicals using rational exponents. The example can be rewritten as follows. x34xx3xx6x34xx13xx16 \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}}}


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.
x34xx13xx16\dfrac{x^{\frac{3}{4}}\cdot x \cdot x^{\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}
x34x1x13xx16\dfrac{x^{\frac{3}{4}}\cdot x^1 \cdot x^{\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}
x34+1+13xx16\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 12.12.
x34+1+13xx16\dfrac{x^{\frac{3}{4}+1+\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}
x912+1+13xx16\dfrac{x^{\frac{9}{12}+1+\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}
x912+1212+13xx16\dfrac{x^{\frac{9}{12}+\frac{12}{12}+\frac{1}{3}}}{|x| \cdot x^{\frac{1}{6}}}
x912+1212+412xx16\dfrac{x^{\frac{9}{12}+\frac{12}{12}+\frac{4}{12}}}{|x| \cdot x^{\frac{1}{6}}}
x2512xx16\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,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.
x2512xx16\dfrac{x^{\frac{25}{12}}}{|x| \cdot x^{\frac{1}{6}}}


Rewrite expression using radicals

When the expression has been simplified completely, it can be rewritten using radicals. x1112x=x1112x \dfrac{x^{\frac{11}{12}}}{|x|}=\dfrac{\sqrt[12]{x^{11}}}{|x|} This is the simplified expression.


Simplify the expression using the properties of exponents. 21034431613 \dfrac{2^{\frac{10}{3}}}{4^{\frac{4}{3}}\cdot 16^{\frac{1}{3}}}

Show Solution
Notice that the terms in this expression each have a difference base. 2,4,and16. 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 2.2. Notice that 4=224=2^2 and 16=24.16=2^4.
21034431613\dfrac{2^{\frac{10}{3}}}{4^{\frac{4}{3}}\cdot 16^{\frac{1}{3}}}
2103(22)431613\dfrac{2^{\frac{10}{3}}}{\left(2^2 \right)^{\frac{4}{3}}\cdot 16^{\frac{1}{3}}}
2103(22)43(24)13\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.
2103(22)43(24)13\dfrac{2^{\frac{10}{3}}}{\left(2^2 \right)^{\frac{4}{3}}\cdot \left(2^4 \right)^{\frac{1}{3}}}
210322432413\dfrac{2^{\frac{10}{3}}}{2^{2 \cdot \frac{4}{3}}\cdot 2^{4 \cdot \frac{1}{3}}}
2103283243\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.
2103283243\dfrac{2^{\frac{10}{3}}}{2^{\frac{8}{3}}\cdot 2^{\frac{4}{3}}}
2-232^{\text{-} \frac{2}{3}}
The answer is 2-23,2^{\text{-} \frac{2}{3}}, which can also be written as 1223. \dfrac{1}{2^{\frac{2}{3}}}.
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