Question

find all the cube roots of -729.

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the real cube root(s) of -729 is/are .
(simplify your answer. use a comma to separate answers as needed.)

b. there are no real cube roots of -729.

Explanation:

Step1: Recall the definition of cube root

The cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \). For real numbers, we know that the cube of a negative number is negative, and the cube of a positive number is positive. So, we are looking for a real number \( y \) such that \( y^3=-729 \).

Step2: Find the number whose cube is -729

We know that \( 9^3 = 9\times9\times9 = 729 \). Then, \((-9)^3=(-9)\times(-9)\times(-9)= -729\) (because the product of two negative numbers is positive, and the product of a positive and a negative number is negative: \((-9)\times(-9) = 81\), then \(81\times(-9)=-729\)).

Answer:

A. The real cube root(s) of \(-729\) is/are \(-9\).