Chapter 7, Problem 34CR

To determine

To fill: The blank and complete the equation ${\left(a^{\frac{1}{2}}\right)}^{?}=a^{\frac{1}{4}}$ .

Answer to Problem 34CR

The equation is ${\left(a^{\frac{1}{2}}\right)}^{\frac{1}{2}}=a^{\frac{1}{4}}$ .

Explanation of Solution

Given information:

Theequation is ${\left(a^{\frac{1}{2}}\right)}^{?}=a^{\frac{1}{4}}$ .

Calculation:

Consider the equation as ${\left(a^{\frac{1}{2}}\right)}^x=a^{\frac{1}{4}}$ .

Simplify the equation ${\left(a^{\frac{1}{2}}\right)}^x=a^{\frac{1}{4}}$ to obtain the value of xas follows:

  $\begin{array}{c}{\left(a^{\frac{1}{2}}\right)}^x=a^{\frac{1}{4}}\\ a^{\frac{1}{2}x}=a^{\frac{1}{4}}\end{array}$

Compare the exponentson both sides of the equation.

  $\begin{array}{c}\frac{1}{2}x=\frac{1}{4}\\ x=\frac{1}{4}\times 2\\ x=\frac{1}{2}\end{array}$

Therefore, the value of xis $\frac{1}{2}$ .

Thus, the equation is ${\left(a^{\frac{1}{2}}\right)}^{\frac{1}{2}}=a^{\frac{1}{4}}$ .