Chapter 6, Problem 7CR

To determine

To Evaluate: The expression ${\left(625\right)}^{\frac{3}{4}}$

Answer to Problem 7CR

The value of ${\left(625\right)}^{\frac{3}{4}}=125$

Explanation of Solution

Given information:

The expression ${\left(625\right)}^{\frac{3}{4}}$

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

  $\begin{array}{l}{\left(625\right)}^{\frac{3}{4}}\\ \text{Factor 625}\\ {\left(625\right)}^{\frac{3}{4}}={\left[5\times 5\times 5\times 5\right]}^{\frac{3}{4}}\\ {\left(625\right)}^{\frac{3}{4}}={\left[{\left(5\right)}^4\right]}^{\frac{3}{4}}\\ \text{Using }{\left(a^m\right)}^n=a^{mn}\\ {\left(625\right)}^{\frac{3}{4}}={\left[5\right]}^{4\times \frac{3}{4}}\\ {\left(625\right)}^{\frac{3}{4}}={\left(5\right)}^{1\times 3}\\ {\left(625\right)}^{\frac{3}{4}}=5^3\\ {\left(625\right)}^{\frac{3}{4}}=125\end{array}$

Conclusion:

The value of ${\left(625\right)}^{\frac{3}{4}}=125$