Question

What are all the cube roots of -64?

Answer 1

Hint: We need to find the cube roots of -64. For this we need to understand the meaning of cube root. For any number, the cube root is the number raised to the power $\dfrac{1}{3}$. Or in other words, any number that when multiplied by three times will give back the original number is called the cube root of that number.

Complete step-by-step answer:
If we have some number and after multiplying it by itself three times we get a new number, then the number obtained is called the ‘cube’ of the value multiplied and the number which was multiplied three times is called the ‘cube root’ of the value obtained after multiplying. In mathematical terms,
For any number $a$, we say that $b$ is the cube root of $a$ if:
$a=b^3$
In other words:
$b=a^{\dfrac{1}{3}}$
$a$ is the cube of $b$ and $b$ is the cube root of $a$.
For example:
$8=2\times 2\times 2$
So we can say that 2 is the cube root of 8 and 8 is the cube of 2.
We are given -64. We see that:
$64=4\times 4\times 4$
But we have a negative sign here as well so we reformulate the above equation as:
$-64=-4\times -4\times -4$
Hence, the cube root of -64 is -4.

Note: Do not get confused between the terms cube and cube root. Here, the cube root has been asked so do not multiply -64 with itself three times because that would result in the cube of the number whereas we are required to find the cube root. Moreover, do take care of the negative sign involved.