Cubes and Cube Roots
To cube a number, use it in multiplication three times.
For, example, cube of $$$5$$$ is $$${5}\times{5}\times{5}={125}$$$.
When we talked about exponents and integers, we said that the number $$$a$$$ raised to $$$b$$$-th power is the number $$$a$$$ multiplied by itself $$$b$$$ times: $$$\color{purple}{a^b=\underbrace{a\cdot a\cdot a\cdot a\cdot...\cdot a}_{b}}$$$.
So, to cube means to raise to the third power.
Thus, we can write, that the cube of the number $$$5$$$ is
$$$5^{\color{purple}{3}}=125$$$
It is read as "5 cubed equals 125".
We can cube any numbers: negative numbers, fractions, decimals, etc.
Just a couple of fast examples, because we already did this (see exponents and integers, negative exponents): $$${{\left(-{3}\right)}}^{{3}}=-{27}$$$, $$${{\left(\frac{{1}}{{2}}\right)}}^{{3}}=\frac{{1}}{{8}}$$$, $$${{\left({0.7}\right)}}^{{2}}={0.343}$$$.
Remember, that raising to the power is just multiplying certain number of times, and cubing is raising to the third power.
Now, suppose we want to do reverse operation. We are given number, and want to find a value, that when cubed will give initial number. This value is called cube root of the number.
Cube root of the number $$${b}$$$ is such number $$${a}$$$, that $$${{a}}^{{3}}={b}$$$.
Cube root of the number has special notation: $$$\color{purple}{\sqrt[3]{b}}$$$.
For, example since we know, that $$${5}\times{5}\times{5}={125}$$$, then cube root of $$${125}$$$ is $$${5}$$$: $$${\sqrt[{{3}}]{{{125}}}}={5}$$$.
Following table contains cubes and cube roots of first 10 whole numbers:
| Number $$${a}$$$ (cube root) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Number $$${b}$$$ (cube of a number) | 0 | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
Highlighted numbers are numbers from above example.
Looking at this, we can easily see, that $$${{7}}^{{3}}={343}$$$ and $$${\sqrt[{{3}}]{{{343}}}}={7}$$$.
Cube root symbol $$$\color{purple}{\sqrt[3]{\phantom{0}}}$$$ is the radical with a small 3 (radical without number is used to write square roots).
This is done to emphasize the fact, that we are looking for a number, that when cubed (i.e. raised to 3-rd power) will give original number.
Finally, notice that cube root undoes cubing (if we take cube root of a number and then cube the result, we will get back to the original number), and vice versa:
$$${\color{red}{{{{\left({\sqrt[{{3}}]{{{b}}}}\right)}}^{{3}}={b}}}}$$$
$$${\color{green}{{{\sqrt[{{3}}]{{{{b}}^{{3}}}}}={b}}}}$$$
Let's go through a couple of example.
Example 1. $$${{\left({\sqrt[{{3}}]{{{64}}}}\right)}}^{{3}}={{4}}^{{3}}={64}$$$.
Example 2. $$${{\left({\sqrt[{{3}}]{{{15}}}}\right)}}^{{3}}={15}$$$.
Example 3. $$${{\left({\sqrt[{{3}}]{{-{5}}}}\right)}}^{{3}}=-{5}$$$.
Example 4. $$${\sqrt[{{3}}]{{{{5}}^{{3}}}}}={\sqrt[{{3}}]{{{125}}}}={5}$$$.
Example 5. $$${\sqrt[{{3}}]{{{{\left(-{2}\right)}}^{{3}}}}}={\sqrt[{{3}}]{{-{8}}}}=-{2}$$$.