Intro to cube roots

The cube root of $y$, written $\sqrt[3]{y}$ or $y^{1/3}$, is the number that gives $y$ when cubed.
Since $2^3 = 8$, $\sqrt[3]{8} = 2$; since $3^3 = 27$, $\sqrt[3]{27} = 3$.

A cube root undoes cubing, so $\sqrt[3]{a^3} = a$.
This makes it the natural tool when a variable appears cubed.
If $r^3 = 8000$, then $r = \sqrt[3]{8000}$, and since $8000 = 8 \times 1000 = 2^3 \times 10^3$, $r = 2 \times 10 = 20$.

Cube roots behave well with negative numbers, unlike square roots.
Because a negative number cubed is negative, $\sqrt[3]{-8} = -2$.
So $\sqrt[3]{y}$ is defined for every real $y$, positive or negative.

They also connect to solving power equations.
If $c^{mn} = 125$ and $m, n$ are integers, then $c = 5$ because $5^3 = 125$, so $mn = 3$.
Recognising a perfect cube like $125 = 5^3$ turns the equation into a simple comparison of exponents.

What is $\sqrt[3]{64}$?
Ask what number cubed gives $64$; since $4^3 = 64$, the cube root is $4$.
So $\sqrt[3]{64} = 4$.

Solve $m^3 = 1000$.
Take the cube root of both sides: $m = \sqrt[3]{1000}$, and $1000 = 10^3$.
So $m = 10$.

Evaluate $\sqrt[3]{-27}$.
A cube root of a negative number is negative, and $(-3)^3 = -27$.
So $\sqrt[3]{-27} = -3$.