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Square Roots
Tutorial 12c
Introduction to Square Roots

Just as the inverse of addition is subtraction,
and of multiplication is division, the inverse
operation of squaring a number is finding a
square root.

An expression involving a square root such as,
81 , is called a radical expression.

The symbol

It indicates the nonnegative or principal square root of
the radicand. The radicand is the term or expression
inside the radical sign.
For Example: In 81 , 81 is the radicand.

is a radical sign.
Introduction to Square Roots


cont. . .
25 is read “square root of 25”.
The square root of 25 is a number that when squared
equals 25.

25 = ?
2
Try different numbers until you find a
number that when squared, it equals 25.





12
22
32
42
52
=
=
=
=
=
1
4
9
16
25 !

25 = 52
25 is 5 squared.

25  5
So, 5 is a square root of 25.
Simplifying Square Roots
Simplify each expression.
a.
36 
b. - 64 
16
c. 

25
d.
0
e.
- 49 
f.
2
Click here
to check your answers
Simplifying Square Roots
Simplify each expression.
36 
6
positive square root
b. - 64  -8
4
16
c. 

5
25
negative square root
a.
d.
e.
0
0
- 49  Does not exist
The square roots are
4
and  4 .
5
5
There is only one square
root of 0.
For real numbers, the square root
of a negative number is undefined.
Estimating & Using Square Roots
Estimate 53 by determining what two
consecutive integers it lies between.
1st 13 Perfect Squares To estimate radicals, we must become familiar
 12 = 1
with perfect squares. Study the table to the left.
 22 = 4
The numbers on the right side of the table are all
 32 = 9
perfect squares.
 42 = 16
53 is between the two consecutive perfect squares
 62 = 36
49 and 64.
 72 = 49
49  53 
64
 82 = 64
 92 = 81
 102 = 100
7 <
<
8
53
 112 = 121
Therefore, 53 is between 7 and 8.
 122 = 144
 132 = 169
Estimating & Using Square Roots
cont. . .
A calculator can be used to approximate radical
expressions.

Use a calculator to approximate 31 .
To find
.
31, press 3 , 1 , then
Your result should be, 31  5.57 (rounded to the nearest
hundredth)

Use a calculator to approximate the following:
Round your answers to the nearest hundredth.
Click here to check your answers.
1.
5
4. 125
2.
22
5. 160
3.
79
6. - 17
Estimating & Using Square Roots
cont. . .
A calculator can be used to approximate radical
expressions.

Use a calculator to approximate 31 .
To find
.
31, press 3 , 1 , then
Your result should be, 31  5.57 (rounded to the nearest
hundredth)

Use a calculator to approximate the following:
Round your answers to the nearest hundredth.
1.
5
 2.24 2. 22  4.69
4. 125  11.18 5. 160  12.65
3.
79  8.89
6. - 17  -4.12
Using Square Roots to Solve
Equations



You may recall from a previous unit that a quadratic
equation is any equation that can be written in the form:
ax2 + bx + c = 0, where a, b, and c are all real numbers
and a  0.
When b = 0 we have a quadratic that can be solved by
using square roots.
For Example: Find the value(s) of x that satisfies this equation.
Solve 3x2 – 24 = 0
3x2 – 24 + 24 = 0 + 24
3x2 = 24
x2 = 8
x2   8
x 8
x  2.83
Get x alone on one side.
 Add 24 to each side.
 Divide each side by 3.
 To undo the square, take the
square root.
 Use a calculator.
Using Square Roots to Solve
Equations cont. . .
Use square roots to solve each quadratic equation.
Round your answers to the nearest hundredth.
1. 5x2 = 80
2. -10x2 + 90 = 0
3. 4x2 + 18 = 162
Click here
to check your answers
Using Square Roots to Solve
Equations cont. . .
Use square roots to solve each quadratic equation.
Round your answers to the nearest hundredth.
1. 5x2 = 80
x2 = 16
x 2   16
x =4
2. -10x2 + 90 = 0
-10x2 = -90
x2 = 9
3. 4x2 + 18 = 162
4x2 = 180
x2 = 45
x2   9
x 2   45
x =3
x   6.71