Question

how many third roots does -512 have? explain your reasoning.

choose the correct answer below and, if necessary, fill in the answer boxes to complete your choice.

a. there is 1 real root, which is
. this times itself
times equals -512.

b. there are 2 real roots, which are
. each of these times themselves
times equals -512.
(use a comma to separate answers as needed.)

c. there are 3 real roots, which are
. each of these times themselves
times equals -512.
(use a comma to separate answers as needed.)

d. there are no real roots. there is no real number that when multiplied by itself any number of times will equal -512.

Explanation:

Step1: Recall the property of cube roots

For any real number \( a \), the equation \( x^3=a \) has exactly one real solution (since the cube function \( f(x) = x^3 \) is a one - to - one function over the real numbers. That is, if \( x_1^3=x_2^3 \), then \( x_1 = x_2 \)).

Step2: Find the real cube root of - 512

We know that \( (-8)^3=(-8)\times(-8)\times(-8)=64\times(-8)= - 512 \). So the real cube root of - 512 is - 8. And we need to multiply - 8 by itself 3 times to get - 512.

Answer:

A. There is 1 real root, which is \(-8\). This times itself \(3\) times equals \(-512\).