Chapter 5.3, Problem 3Q

a.

To determine

To calculate:

The value of ${16}^{\frac{3}{4}}$

Answer to Problem 3Q

The value of ${16}^{\frac{3}{4}}$ is 8

Explanation of Solution

Given information:

Calculation:

The value of ${16}^{\frac{3}{4}}$ can be calculated as,

Firstly find out the factor of 16.

So, the factor of 16 is $2\times 2\times 2\times 2=2^4$

Now,

  $\begin{array}{l}{16}^{\frac{3}{4}}={\left(2^4\right)}^{\frac{3}{4}}\\ {16}^{\frac{3}{4}}={\left(2\right)}^{4\times \frac{3}{4}}\left[\text{Apply }{\left(a^m\right)}^n=a^{mn}\right]\\ {16}^{\frac{3}{4}}={\left(2\right)}^3\\ {16}^{\frac{3}{4}}=8\end{array}$

Therefore, the value of ${16}^{\frac{3}{4}}$ is 8

b.

To determine

To calculate:

The value of ${125}^{\frac{2}{3}}$

Answer to Problem 3Q

The value of ${125}^{\frac{2}{3}}$ is 25

Explanation of Solution

Given information:

Calculation:

The value of ${125}^{\frac{2}{3}}$ can be calculated as,

Firstly find out the factor of 125.

So, the factor of 125 is $5\times 5\times 5=5^3$

Now,

  $\begin{array}{l}{125}^{\frac{2}{3}}={\left(5^3\right)}^{\frac{2}{3}}\\ {125}^{\frac{2}{3}}={\left(5\right)}^{3\times \frac{2}{3}}\left[\text{Apply }{\left(a^m\right)}^n=a^{mn}\right]\\ {125}^{\frac{2}{3}}={\left(5\right)}^2\\ {125}^{\frac{2}{3}}=25\end{array}$

Therefore, the value of

  ${125}^{\frac{2}{3}}$ is 25