Chapter 6, Problem 6CR

To determine

To Evaluate: The expression $\sqrt[5]{-243}$

Answer to Problem 6CR

The value of $\sqrt[5]{-243}=-3$

Explanation of Solution

Given information:

The expression $\sqrt[5]{-243}$

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

  $\begin{array}{l}\sqrt[5]{-243}\\ \text{Factor 243}\\ \sqrt[5]{-243}=\sqrt[5]{\left(-3\right)\times \left(-3\right)\times \left(-3\right)\times \left(-3\right)\times \left(-3\right)}\\ \text{Using }\sqrt[n]{a}={\left(a\right)}^{\frac{1}{n}}\\ \sqrt[5]{-243}={\left[{\left(-3\right)}^5\right]}^{\frac{1}{5}}\\ \text{Using }{\left(a^m\right)}^n=a^{mn}\\ \sqrt[5]{-243}={\left(-3\right)}^{5\times \frac{1}{5}}\\ \sqrt[5]{-243}={\left(-3\right)}^1\\ \sqrt[5]{-243}=-3\end{array}$

Conclusion:

The value of $\sqrt[5]{-243}=-3$