Chapter 6.4, Problem 7Q

To determine

To Evaluate: The expression ${\left(512\right)}^{\frac{2}{3}}$

Answer to Problem 7Q

The value of ${\left(512\right)}^{\frac{2}{3}}=64$

Explanation of Solution

Given information:

The expression ${\left(512\right)}^{\frac{2}{3}}$

Concept and Formula Used:

Rule of exponent and addition and subtraction.

  $a^n=a\times a\times a\times \mathrm{..................}n\text{ times,}\sqrt[n]{a}={\left(a\right)}^{\frac{1}{n}}\&{\left(a^m\right)}^n=a^{mn}$

Calculation:

The given expression is

  $\begin{array}{l}{\left(512\right)}^{\frac{2}{3}}\\ \text{Factor 512}\\ {\left(512\right)}^{\frac{2}{3}}={\left[2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\right]}^{\frac{2}{3}}\\ {\left(512\right)}^{\frac{2}{3}}={\left[{\left(2\right)}^9\right]}^{\frac{2}{3}}\\ \text{Using }{\left(a^m\right)}^n=a^{mn}\\ {\left(512\right)}^{\frac{2}{3}}={\left[2\right]}^{9\times \frac{2}{3}}\\ {\left(512\right)}^{\frac{2}{3}}={\left(2\right)}^{3\times 2}\\ {\left(512\right)}^{\frac{2}{3}}=2^6\\ {\left(512\right)}^{\frac{2}{3}}=64\end{array}$

Conclusion:

The value of ${\left(512\right)}^{\frac{2}{3}}=64$