What is a root?
Let's start with the most common root, the square root. The square root of a number can be multiplied by itself to obtain the original number. A more formal definition is shown below. The number k is the n-th root of the number y.
kn=yExample
The principal square root of 4 is 2. Why? Because 2∗2=4. Since we are multiplying it by itself once, we have two copies of the number -- 2∗2=4. Similarly, 3∗3=9, so 3 is the square root of 9.
Special Cases
What is the principal square root of 1? It's 1. That's actually pretty obvious -- 1∗1=1 so it works just like normal.
What is the square root of 0? Same thing -- 0∗0=0.
What is the square root of -4? There is none! There is no number that you can multiply by itself to get -4. After all, if you multiply a positive number by itself you get a positive, and if you multiply two negative numbers you STILL get a positive. There's no way to square a number and get a negative.
Can a number have more than one root?
Yes! Notice very carefully that I used the term principal square root above. I was trying to avoid getting too complicated too fast. In fact, the number 4 has two square roots. They are +2 and -2. Try it out. Multiply either of them by itself and you still get 4. Often we are just looking for the positive root, so you'll often hear the term square root used to mean the positive root, or the principal root, but do keep in mind that a number may have more than one square root.
What is a cube root?
A cube root, or cubic root, is similar to a square root except it is multiplied by itself three times. For example, 2 is the cubic root of 8 because 2∗2∗2=8. A negative number can have a cubic root: The cubic root of -27 is -3, because −3∗−3∗−3=−27.
What does nth root mean?
We can extend beyond square and cube roots and keep going, which is what nth root means. The n would stand for some number, like 5. The 5th root of 32 is 2, because 2∗2∗2∗2∗2=32.