Chapter 7.9, Problem 71WE

To determine

To calculate: The simplified form of the expressions ${\left(x^2\right)}^{\frac{1}{2}}$ and ${\left(x^3\right)}^{\frac{1}{3}}$ .

Answer to Problem 71WE

The simplified form of the expression ${\left(x^2\right)}^{\frac{1}{2}}$ is $x$ and ${\left(x^3\right)}^{\frac{1}{3}}$ is $x$ .

Explanation of Solution

Given information:

The expressions ${\left(x^2\right)}^{\frac{1}{2}}$ and ${\left(x^3\right)}^{\frac{1}{3}}$ .

Formula used:

Law of exponent for power of power is ${\left(a^m\right)}^n=a^{mn}$ where a is a nonzero integers and n and m are positive integers.

Calculation:

Consider the expression ${\left(x^2\right)}^{\frac{1}{2}}$ .

Recall that law of exponent for power of power is ${\left(a^m\right)}^n=a^{mn}$ where a is a nonzero integers and n and m are positive integers.

Apply it,

  $\begin{array}{c}{\left(x^2\right)}^{\frac{1}{2}}=x^{2\left(\frac{1}{2}\right)}\\ =x^1\\ =x\end{array}$

Consider the expression ${\left(x^3\right)}^{\frac{1}{3}}$ .

Recall that law of exponent for power of power is ${\left(a^m\right)}^n=a^{mn}$ where a is a nonzero integers and n and m are positive integers.

Apply it,

  $\begin{array}{c}{\left(x^3\right)}^{\frac{1}{3}}=x^{3\left(\frac{1}{3}\right)}\\ =x^1\\ =x\end{array}$

Thus, the simplified form of the expression ${\left(x^2\right)}^{\frac{1}{2}}$ is $x$ and ${\left(x^3\right)}^{\frac{1}{3}}$ is $x$ .