Chapter 3.1, Problem 3GP

Solution

The nth root(s) of $a=-64$ .

  $-4$

Given:

The number $a=-64,\text{ and }n=3$ .

Concept Used:

• The even inter root of a number is defined for positive values only that are greater than or equal to 0.
• The odd integer root of all real numbers is defined.
• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$

Calculation:

In order to find the 3^rd root of the number $a=-64$ , note that the number is positive and the n is an odd integer, so there must be one real root of the number.

To find the 3^rd roots of the given number, rewrite the number in factored form and then simplify further using the exponent properties as shown below

  $\begin{array}{c}\sqrt[3]{-64}={\left(-64\right)}^{\frac{1}{3}}\\ ={\left({\left(-4\right)}^3\right)}^{\frac{1}{3}}\\ ={\left(-4\right)}^{3\cdot \frac{1}{3}}\\ ={\left(-4\right)}^1\\ =-4\end{array}$

Therefore, the 3^rd root of the given number is $-4$ .