10.2 Roots

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 Square root of four, the answer is 2, since equation sequence two multiplication two equals four equals two squared.

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: four super one divided by two multiplication four super one divided by two.

Adding the powers gives: equation sequence four super one divided by two multiplication four super one divided by two equals four super one equals four.

This shows that when you multiply four super one divided by two by itself, you get 4. In other words, four super one divided by two is a square root of 4.

This is defined as the positive square root. It may be expressed as:

equation sequence four super one divided by two equals Square root of four equals two

Similarly, 27 super one divided by three means the cube root of 27, i.e., the number which when multiplied by itself three times gives 27. Since three multiplication three multiplication three equals 27, we say that 3 is the cube root of 27 or 27 super one divided by three equals three.

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 four multiplication four equals 16. But there is another square root of 16 because four minus multiplication four minus equals 16, 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 equation left hand side prefix plus minus of Square root of 16 equals right hand side prefix plus minus of four. The symbol ± is read as “plus or minus.” The Square root of symbol means “the positive square root of … .”

So, we can say that the square roots of 25 are prefix plus minus of five, while Square root of 25 equals five.

10.1.11 Decimal Forms of Powers of Ten and Prefixes

10.2.1 Calculator Exploration: Square Roots