Chapter 3.1, Problem 1GP

Solution

The nth root(s) of $a=625$ .

  $\pm 5$

Given:

The number $a=625,\text{ and }n=4$ .

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 4^th root of the number $a=625$ , note that the number is positive and the n is an even integer, so there must be two real 4^th roots of the number.

To find the 4^th roots of the given number, rewrite the number in factored form and then simplify further using the exponent properties as shown below

  $\begin{array}{c}\sqrt[4]{625}=\pm {\left(625\right)}^{\frac{1}{4}}\\ =\pm {\left(5^4\right)}^{\frac{1}{4}}\\ =\pm 5^{4\cdot \frac{1}{4}}\\ =\pm 5^{\frac{4}{4}}\\ =\pm 5^1\\ =\pm 5\end{array}$

Therefore, the 4^th roots of the given number are $\pm 5$ .