Chapter 3.1, Problem 4GP

Solution

The nth root(s) of $a=243$ .

  $3$

Given:

The number $a=243,\text{ and }n=5$ .

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 5^th root of the number $a=243$ , 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 5^th roots of the given number, rewrite the number in factored form and then simplify further using the exponent properties as shown below

  $\begin{array}{c}\sqrt[5]{243}={\left(243\right)}^{\frac{1}{5}}\\ ={\left(3^5\right)}^{\frac{1}{5}}\\ =3^{5\cdot \frac{1}{5}}\\ =3^1\\ =3\end{array}$

Therefore, the 5^th root of the given number is $3$ .