Chapter 7.3, Problem 12CYU

To determine

To simplify the given expression.

Answer to Problem 12CYU

  $\frac{1}{343}$

Explanation of Solution

Given:

Expression: ${\left(\frac{1}{49}\right)}^{\frac{3}{2}}$

Formula used:

  $x^{\frac{a}{b}}=\sqrt[b]{x^a}$

Calculation:

Consider the given expression ${\left(\frac{1}{49}\right)}^{\frac{3}{2}}$ ,

Applying the above formula,

  ${\left(\frac{1}{49}\right)}^{\frac{3}{2}}=\sqrt{{\left(\frac{1}{49}\right)}^3}$

Here, $\frac{1}{49}$ can be written as ${49}^{-1}$ and 49 can be written as $7^2.$

  $\begin{array}{c}{\left(\frac{1}{49}\right)}^{\frac{3}{2}}=\sqrt{{\left(\frac{1}{49}\right)}^3}\\ =\sqrt{{49}^{-3}}\\ =\sqrt{{\left(7^2\right)}^{-3}}\\ =7^{\frac{2\times -3}{2}}\\ =7^{-3}\\ =\frac{1}{343}\end{array}$

Conclusion:

Therefore, the simplified form of the expression is $\frac{1}{343}$ .