Question

find the root of (sqrt5{-243}). select the correct choice below and, if necessary, fill in the answer box to complete your choice. (\bigcirc) a. (sqrt5{-243} = square) (simplify your answer.) (\bigcirc) b. the root is not a real number.

Explanation:

Step1: Recall the property of odd roots

For an odd - numbered root \( \sqrt[n]{a} \), if \( n \) is odd, then \( \sqrt[n]{-a}=-\sqrt[n]{a} \). Here, \( n = 5 \) (which is odd) and we need to find \( \sqrt[5]{- 243} \).

Step2: Find the fifth root of 243

We know that we need to find a number \( x \) such that \( x^{5}=-243 \). Let's first consider the positive case: find \( y \) such that \( y^{5}=243 \). We know that \( 3^{5}=3\times3\times3\times3\times3 = 243 \).

Step3: Apply the odd - root property

Since \( n = 5 \) (odd), \( \sqrt[5]{-243}=-\sqrt[5]{243} \). And since \( \sqrt[5]{243} = 3 \) (because \( 3^{5}=243 \)), then \( \sqrt[5]{-243}=-3 \).

Answer:

A. \( \sqrt[5]{-243}=\boxed{-3} \)