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.
Note: You can not take the square root of a negative number.
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 irrational
number, so 121 is not an irrational number.
The answer is A.
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
4
.
9
example 4
Simplify
32
.
20
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
A.
B.
C.
D.
72 ?
7.2
9.1
8.9
8.5
example 6
The square root of 31 is between which two whole numbers?
A.
B.
C.
D.
4 and 5
5 and 6
6 and 7
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.
Q
0
5
R
S
10
15
T
20
25
Which letter best represents the location of 50 ?
A.
B.
C.
D.
Q
R
S
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.
Name ______________________________
Simplify. Leave in radical form (in other words, don’t use a calculator).
1)
20
2) − 64
3)
900
4)
0.16
5) − 0.0025
6)
16
25
7)
3
27
8)
10)
8 · 32
4
3
11) 5 14 · 2 7
9)
3
7
12) − 3 5 ·
4
5
Name the integers between which each value lies (without using a calculator).
13)
11
15) 2 3
17)
25
2
14 )
34
16 ) 3 6
18)
14
2
Label each statement as true or false (without using a calculator).
19) 2 < 7 < 3
20) 62 < 65 < 72
21) 22 < 20 < 32
22)
45 < 10 < 90