In Unit 4 we started to look at exponents and looked at multiplying numbers by themselves a number of times, as we have been looking at with powers of ten in this unit. The root of a number reverses this process. So if you take the square root of 4, written as , the answer is 2, since .
Since a number can be multiplied by itself any number of times there are also any number of different roots as well. We will look at this further later in this unit.
You can express square roots using power notation as well. Consider the following: .
Adding the powers gives: .
This shows that when you multiply by itself, you get 4. In other words, is a square root of 4.
This is defined as the positive square root. It may be expressed as:
Similarly, means the cube root of 27, i.e., the number which when multiplied by itself three times gives 27. Since , we say that 3 is the cube root of 27 or .
Convince yourself that this is the case by using the rules of multiplying powers with the same base.
A square root of a number, say 16, is a value that when multiplied by itself, gives the number (16 here). So one square root of 16 is 4 because . But there is another square root of 16 because , too. Any positive number has two square roots, a positive one and a negative one.
These square roots can be written down using the square root notation: The square roots of 16 are . The symbol ± is read as “plus or minus.” The symbol means “the positive square root of … .”
So, we can say that the square roots of 25 are , while .