Q. 8.62 - Intermediate Algebra

Question

Simplify:

(a) $4^{\frac{3}{2}}$

(b) ${27}^{-\frac{2}{3}}$

(c) ${625}^{-\frac{3}{4}}$

Step-by-Step Solution

Answer

The simplification of $4^{\frac{3}{2}}$ is $8$

The simplification of ${27}^{-\frac{2}{3}}$ is $\frac{1}{9}$

The simplification of ${625}^{-\frac{3}{4}}$ is $\frac{1}{125}$

Part (a) Step 1. Given

An expression $4^{\frac{3}{2}}$

To simplify the given expression

Part (a) Step 2. Change to radical form

The power of the radical is the numerator of the exponent, $3$.
The index of the radical is the denominator of the exponent, $2$.

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

      $=({\sqrt{\left(2^2\right)})}^3$

      $=2^3$

     $=8$

Part (b) Step 1. Given

An expression ${27}^{-\frac{2}{3}}$

To simplify the given expression

Part (b) Step 2. Change to radical form

By using the property, $a^{-n}=\frac{1}{a^n}$,

${27}^{-\frac{2}{3}}=\frac{1}{{27}^{{\displaystyle \frac{2}{3}}}}$

The power of the radical is the numerator of the exponent, $2$.
The index of the radical is the denominator of the exponent, $3$

${27}^{-\frac{2}{3}}=\frac{1}{{\left(\sqrt[3]{27}\right)}^2}$

          $=\frac{1}{(\sqrt[3]{3^3{)}^2}}$

          $=\frac{1}{3^2}$

          $=\frac{1}{9}$

Part (c) Step 1. Given

An expression ${625}^{-\frac{3}{4}}$

To simplify the given expression

Part (c) Step 2. Change to radical form

By using the property, $a^{-n}=\frac{1}{a^n}$,

${625}^{-\frac{3}{4}}=\frac{1}{{625}^{{\displaystyle \frac{3}{4}}}}$

The power of the radical is the numerator of the exponent, $3$.
The index of the radical is the denominator of the exponent, $4$.

${625}^{-\frac{3}{4}}=\frac{1}{{\left(\sqrt[4]{625}\right)}^3}$

            $=\frac{1}{{\left(\sqrt[4]{5^4}\right)}^3}$

           $=\frac{1}{5^3}$

          $=\frac{1}{125}$