Square Roots

2a b

• a is a square root of b if and only if

• Example 1

3 is a square root of 9, since 23 9

- 3 is also a square root of 9, since 23 9

Thus, 9 has two square roots, -3 and 3.

• Example 2

The number 0 has only one square root, which is 0 since

20 0

• Sometimes there are no square roots of a number.

• Example 3:

There are no square roots of –16 since there are no numbers a such that

216a

24 16 2

4 16

Note that neither 4 nor – 4 will work.

a

The expression that appears under the radical sign, a in this case, is called the radicand.

• Radical notation

Radical symbol

Square root of a

When radical notation is used, the result is the principal square root, which is the non-negative root.

• Example 4

25Simplify:

There are two square roots of 25, - 5 and 5.

The principal second root is the non-negative root, or 5. Therefore,

25 5

• Example 5

9 3

Simplify Reasoning

23 9

100 10 210 100

Note that while it is also true that 210 100

we want the principal (nonnegative) root.

• Example 6

Simplify Reasoning

4 2

4 1 4

1 2

2

• Example 7

Simplify Reasoning

4 ?

2

2

2 4

2 4

is undefined.4

• Example 8

9 3

25 5

Simplify Reasoning

23 9

5 25