Understanding Square Roots

The square root of a given number is a value that, when multiplied by itself, produces that given number. Square roots come in pairs, a positive root and a negative root.

For example, a square root of 4 is 2, because 2 × 2 = 4. Another square root of 4 is −2,because −2 × −2 = 4. This can be written as 4 = ±2, which means that 4 = 2 and

4 = −2.

To better understand square roots, it is very helpful to be familiar with perfect squares. A perfect square is the square of an integer:

example 1

Which of the following statements about 121 is not true?

A. 121 is an irrational number. B. 121 is an integer. C. 121 is a real number.

D. 121 is a rational number.

To answer this question, knowledge of perfect squares is extremely helpful. A square root is an irrational number, unless it is the square root of a perfect square:

11 is an integer and a real number and a rational number, but 11 is not an irrationalnumber, so 121 is not an irrational number.

The answer is A.

Note: You can not take the square root of a negative number.

Simplifying Square Roots

To simplify the square root of a number, find two factors of that number, one of which is a perfect square.

(A factor is an integer that divides evenly into another number.)

example 2

Simplify 18 .

This example demonstrates the following property of square roots:

This applies to fractions as well:

example 3

Simplify 94 .

example 4

Simplify 2032 .

It’s not acceptable to leave a square root (also called a radical) in the denominator of a fraction. The way to leave an answer without a radical in the denominator is to multiply the numerator and denominator by the same radical that is in the denominator.

This is called rationalizing the denominator.

Estimating the Value of a Square Root

Perfect squares can also be used to estimate the approximate value of a square root.

example 5

Which is the best approximation of 72 ?

A. 7.2 B. 9.1 C. 8.9 D. 8.5

example 6

The square root of 31 is between which two whole numbers?

A. 4 and 5 B. 5 and 6 C. 6 and 7 D. 7 and 8

This problem can also be solved by first making a list of perfect squares.

31 is between 25 and 36, so the square root of 31 is between 25 and 36 , or between 5 and 6.

The answer is B.

example 7

A part of the real number line is shown below.

Which letter best represents the location of 50 ?

A. Q B. R C. S D. T

Once again, a list of perfect squares will help, this time near 50.

50 is really close to 49, so 50 will be close to 49 , or 7. On the number line above, only one point looks like it could represent a value near 7.

The answer is B.

0 5 10 15 20 25

Q R S T

Name ______________________________

Simplify. Leave in radical form (in other words, don’t use a calculator).

1) 20 2) 64− 3) 900

4) 16.0 5) 0025.0− 6) 2516

7)273 8)

34 9)

73

10) 8 · 32 11) 145 · 72 12) 53− ·54

Name the integers between which each value lies (without using a calculator).

13) 11 )41 34

15) 3 )61 2 63

17)225 )81

214

Label each statement as true or false (without using a calculator).

19) 2 < 6 )02 3 <7 2 < 65 < 72

21) 22 < 20 < 32 )22 45 < 10 < 90