Question

find all the cube roots of -729.
select the correct choice below and, if necessary, fill in the answer box to complete your
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 negative numbers, we know that \( (-a)^3=-a^3 \), so negative numbers have real cube roots.

Step2: Find the cube root of -729

We need to find a number \( y \) such that \( y^3=-729 \). Let's check \( (-9)^3 \). We know that \( (-9)\times(-9)=81 \), and \( 81\times(-9)= - 729 \). So \( (-9)^3=-729 \), which means the cube root of -729 is -9. Also, in the complex number system, there are three cube roots, but for real cube roots, we have one real cube root for a negative number (since the cube function is one - to - one over the real numbers: if \( y_1^3=y_2^3 \), then \( y_1 = y_2 \) when dealing with real numbers). But in terms of real cube roots, we have \( \sqrt[3]{-729}=-9 \) because \( (-9)^3=-729 \).

Answer:

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