Chapter 3.1, Problem 18E

Solution

The nth root(s) of $a=343$ .

  $7$

Given:

The number $a=343,\text{ and }n=3$ .

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 $a=343$ , 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

  $\begin{array}{c}\sqrt[3]{343}={\left(343\right)}^{\frac{1}{3}}\\ ={\left(7^3\right)}^{\frac{1}{3}}\\ =7^{3\cdot \frac{1}{3}}\\ =7^1\\ =7\end{array}$

Therefore, the 3^rd root of the given number is $7$ .