Fractional Exponents – Rule and Examples

Algebraic expressions with fractional exponents can be simplified and solved using the fractional exponents’ rule, which relates exponents to radicals. In this article, we will look at the fractional exponent rule. We will use this rule along with the negative exponents’ rule to solve more complex problems.

We will look at various problems with answers to understand the rules fully.

ALGEBRA
fractional exponents rules and exercises

Relevant for

Learning to simplify expressions with fractional exponents.

See examples

ALGEBRA
fractional exponents rules and exercises

Relevant for

Learning to simplify expressions with fractional exponents.

See examples

How to simplify expressions with fractional exponents?

To simplify and solve an expression with a fractional exponent, we have to use the fractional exponent rule, which relates the powers to the roots. The general form of the fractional exponent rule is

law of fractional exponents

Let’s define some terms of this expression:

  • Radicand: The radicand is the expression under the radical sign √. In this case, the radicand is $latex {{b}^m}$.
  • Radical index: The index is the number that indicates which root is being applied. In the expression above, the index is n.
  • Base: The base is the number or variable that is raised to a power. In this case, the base is b.
  • Exponent: The exponent determines how many times the base is being multiplied by itself. In the expression above, the exponent is m/n.

Let’s look at how to solve expressions with fractional exponents with the following examples:

EXAMPLES

  • Simplify the expression $latex {{16}^{{\frac{1}{2}}}}$.

Solution: Applying the fractional exponents rule, we have:

$latex {{16}^{{\frac{1}{2}}}}=\sqrt{{16}}$

$latex =4$

  • Simplify the expression $latex {{4}^{{\frac{3}{2}}}}$.

Solution: Applying the fractional exponents rule, we have:

$latex {{4}^{{\frac{3}{2}}}}=\sqrt{{{{4}^{3}}}}$

Now, we can apply the exponent to the expression that is inside the square root:

$latex =\sqrt{{4\times 4\times 4}}$

$latex =\sqrt{{64}}$

$latex =8$

  • Simplify the expression $latex {{\left( {\frac{8}{{27}}} \right)}^{{\frac{4}{3}}}}$.

Solution: In this case, we can solve this problem in a different way. We can see that the 8 can be rewritten as $latex {{2}^3}$ and the 27 can be rewritten as $latex {{3}^3}$:

$latex {{\left( {\frac{8}{{27}}} \right)}^{{\frac{4}{3}}}}={{\left( {\frac{{{{2}^{3}}}}{{{{3}^{3}}}}} \right)}^{{\frac{4}{3}}}}$

Now, we can combine the fraction and cube the entire fraction to then simplify

$latex ={{\left[ {{{{\left( {\frac{2}{3}} \right)}}^{3}}} \right]}^{{\frac{4}{3}}}}$

$latex ={{\left( {\frac{2}{3}} \right)}^{4}}$

$latex =\frac{{16}}{{81}}$


Radical form to fractional exponent

To transform from radical form to fractional exponent, we have to use the fractional exponent rule inversely. We can form a fractional exponent where the numerator is the exponent to which the base is raised and the denominator is the index of the radical. That is, we use the following relationship:

transform from radical to fractional exponent

EXAMPLES

  • Transform the expression $latex \sqrt[3]{{{{x}^{2}}}}$ to an expression with fractional exponents.

Solution: We use the fractional exponents’ rule in inverse order:

$latex \sqrt[3]{{{{x}^{2}}}}={{x}^{{\frac{2}{3}}}}$

  • Transform the expression $latex \sqrt{{{{x}^{5}}{{y}^{3}}}}$ to an expression with fractional exponents.

Solution: Again, we can apply the fractional exponents rule in inverse order:

$latex \sqrt{{{{x}^{5}}{{y}^{3}}}}={{x}^{{\frac{5}{2}}}}{{y}^{{\frac{3}{2}}}}$


Negative fractional exponents

When we have negative fractional exponents, we have to apply both the negative exponents’ rule and the fractional exponent’s rule. Generally, the easiest way to solve these types of expressions is to start by applying the rule of negative exponents and then apply the rule of fractional exponents.

Recall that the rule of fractional exponents tells us that a negative exponent can be transformed into a positive one by taking the reciprocal of the base. That is, if the base is in the numerator, we change it to the denominator and if the base is in the denominator, we change it to the numerator.

law of negative exponents

EXAMPLES

  • Simplify the expression $latex \frac{1}{{{{{16}}^{{-\frac{1}{2}}}}}}$.

Solution: We start by applying the negative exponents rule to transform the negative exponent to positive:

$latex \frac{1}{{{{{16}}^{{-\frac{1}{2}}}}}}={{16}^{{\frac{1}{2}}}}$

Now, we can apply the rule of fractional exponents:

$latex =\sqrt{{16}}$

$latex =4$

  • Simplify the expression $$\frac{{{{{27}}^{{-\frac{1}{3}}}}{{y}^{{-\frac{2}{3}}}}~}}{{{{x}^{{-\frac{1}{2}}}}~}}$$

Solution: Again, we start with the negative exponents rule:

$$\frac{{{{{27}}^{{-\frac{1}{3}}}}{{y}^{{-\frac{2}{3}}}}~}}{{{{x}^{{-\frac{1}{2}}}}~}}=\frac{{{{x}^{{\frac{1}{2}}}}}}{{{{{27}}^{{\frac{1}{3}}}}{{y}^{{\frac{2}{3}}}}}}$$

Now that we only have positive exponents, we can apply the rule of fractional exponents to eliminate the exponents:

$$=\frac{{\sqrt{x}}}{{\sqrt[3]{{27}}\sqrt[3]{{{{y}^{2}}}}}}$$

$$=\frac{{\sqrt{x}}}{{3~\sqrt[3]{{{{y}^{2}}}}}}$$


Fractional exponents – Examples with answers

EXAMPLE 1

  • Simplify the expression $latex {{x}^{{\frac{1}{2}}}}{{y}^{{\frac{2}{3}}}}$.

Solution: We simply apply the rule of fractional exponents to form radicals:

$latex {{x}^{{\frac{1}{2}}}}{{y}^{{\frac{2}{3}}}}=\sqrt{x}~\sqrt[3]{{{{y}^{2}}}}$

EXAMPLE 2

  • Simplify the expression $latex {{81}^{{\frac{1}{4}}}}{{x}^{{\frac{1}{2}}}}$

Solution: Again, we just have to apply the rule of fractional exponents to form radicals and then we simplify:

$latex {{81}^{{\frac{1}{4}}}}{{x}^{{\frac{3}{2}}}}=\sqrt[4]{{81}}~\sqrt{{{{x}^{3}}}}$

$latex =3~\sqrt{{{{x}^{3}}}}$

EXAMPLE 3

  • Simplify the expression $latex {{4}^{{-\frac{1}{2}}}}{{x}^{{-\frac{1}{2}}}}$.

Solution: Here, we have negative exponents, so we start by transforming negative exponents to positive using the negative exponents rule:

$latex {{4}^{{-\frac{1}{2}}}}{{x}^{{-\frac{1}{2}}}}=\frac{1}{{{{4}^{{\frac{1}{2}}}}{{x}^{{\frac{1}{2}}}}}}$

Now, we use the fractional exponent rule and simplify:

$latex =\frac{1}{{\sqrt{4}~\sqrt{x}}}$

$latex =\frac{1}{{2~\sqrt{x}}}$

EXAMPLE 4

  • Simplify the expression $$\frac{{{{{16}}^{{-\frac{1}{2}}}}~{{y}^{{-\frac{1}{3}}}}}}{{{{x}^{{-\frac{1}{2}}}}~}}$$

Solution: We have negative exponents, so we start with the negative exponents rule:

$$\frac{{{{{16}}^{{-\frac{1}{2}}}}~{{y}^{{-\frac{1}{3}}}}}}{{{{x}^{{-\frac{1}{2}}}}~}}=\frac{{{{x}^{{\frac{1}{2}}}}~}}{{{{{16}}^{{\frac{1}{2}}}}~{{y}^{{\frac{1}{3}}}}~}}$$

Now, we use the fractional exponent rule and simplify:

$latex =\frac{{\sqrt{x}}}{{\sqrt{{16}}~\sqrt[3]{y}}}$

$latex =\frac{{\sqrt{x}}}{{4~\sqrt[3]{y}}}$

Try solving the following practice problems

Simplfy the expression $latex {{64}^{\frac{1}{3}}}$.

Choose an answer






Simplify the expression $latex {{49}^{\frac{1}{2}}}{{x}^{\frac{1}{3}}}$.

Choose an answer






Simplify the expression $latex {{125}^{-\frac{1}{3}}}$.

Choose an answer







See also

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