Square Roots of a Quantity Squared

2a

• An important form of a square root is:

It would seem that we should write …

2a a

… but as we shall see, this is not always the case.

• Example 1

23 9 3

20 0 0

23 9 3

Note the patterns here.

Same

Same

Opposite in sign

• Recall that the absolute value of a negative number is the opposite of that number.

23 3 3

We now define …

2a a

• Example 2

Simplify

28 8 8

• Example 3

Simplify

22x 2x

Since x + 2 could be negative for certain values of x, we must keep the absolute value sign.

• Example 4

Simplify 12a

26a

First write the radicand as a quantity squared.

12a6a

Since 6a is always nonnegative, the absolute valuesign is not necessary.

6a

• Example 5

Simplify 10a

25a

First write the radicand as a quantity squared.

10a5a

Since 5a would be negative if a were negative, the absolute value sign is necessary.

• Example 6

Simplify 216 40 25x x

Try to create the pattern of 2a

To do this, factor the radicand.

216 40 25x x

24 5x

Since 4x - 5 could be negative for certain values of x, we must keep the absolute value sign.

216 40 25x x

24 5x

4 5x

• Sometimes the directions will include a statement that the values of the variables will be such that the radicand will be nonnegative.

• In this case, the absolute value sign is not necessary.

• Example 7

Simplify the expression, assuming that the variable represents a nonnegative value.

10a

25a10a 5a

Since the variable can’t be negative, the absolute value sign is not necessary.