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Section 10.1
Goal

Simplify square roots.
Key Vocabulary
Radical
 Radicand

Square Roots
Opposite of squaring a number is taking the
square root of a number.
A number b is a square root of a number a if
b2 = a.
In order to find a square root of a, you need
a # that, when squared, equals a.
22 = 4
2
2
The square root of 4 is
2
32 = 9
3
3
The square root of 9 is
3
42 = 16
4
4
The square root of 16 is
4
2
5
= 25
5
5
The square root of 25 is
5
Principal Square Roots
Any positive number has two real square roots, one
positive and one negative, √x and -√x
√4 = 2 and -2, since 22 = 4 and (-2)2 = 4
The principal (positive) square root is noted
as
a
The negative square root is noted as
 a
Radicands
Radical expression is an expression
containing a radical sign.
Radicand is the expression under a radical
sign.
Note that if the radicand of a square root is a
negative number, the radical is NOT a real
number.
Perfect Squares
Square roots of perfect square radicands
simplify to rational numbers (numbers that
can be written as a quotient of integers).
Square roots of numbers that are not perfect
squares (like 7, 10, etc.) are irrational
numbers.
IF REQUESTED, you can find a decimal
approximation for these irrational numbers.
Otherwise, leave them in radical form.
Example 1
Use a Calculator to Find Square Roots
Find the square root of 52. Round your answer to the
nearest tenth. Check that your answer is reasonable.
SOLUTION
Calculator keystrokes
52
or
52
Display
Rounded value
7.21110
52 ≈ 7.2
This is reasonable, because 52 is between the perfect
squares 49 and 64. So, 52 should be between 49 and
64, or 7 and 8. The answer 7.2 is between 7 and 8.
Perfect Squares
The terms of the following sequence:
1, 4, 9, 16, 25, 36, 49, 64, 81…
12,22,32,42, 52 , 62 , 72 , 82 , 92…
These numbers are called the
Perfect Squares.
Properties of Square Roots
Properties of Square Roots (a, b > 0)
Product Property
ab  a  b
18  9  2  3 2
Quotient Property
a
a

b
b
2
2
2


25
5
25
Product Property Example
Use the product property to simplify the
following radical expressions.
40  4  10  2 10
15
No perfect square factor, so
the radical is already
simplified.
Quotient Property Example
Use the quotient property to simplify the
following radical expression.
5

16
5
5

4
16
Sums and Differences
The product and quotient properties allow us
to split radicals that have a radicand which is
a product or a quotient.
We can NOT split sums or differences.
ab  a  b
a b  a  b
Like Radicals
“Like Terms” are terms with the same variables raised to
the same powers.
They can be combined through addition and
subtraction.
Similarly, we can work with the concept of “like” radicals
to combine radicals with the same radicand.
Like radicals are radicals with the same index and the
same radicand.
Like radicals can also be combined with addition or
subtraction by using the distributive property.
Adding and Subtracting Radical Expressions
37 3  8 3
10 2  4 2  6 2
3
24 2
Can not simplify
5 3
Can not simplify
Adding and Subtracting Radical Expressions
Simplify the following radical expression.
 75  12  3 3 
 25  3  4  3  3 3 
 25  3  4  3  3 3 
5 3  2 3  3 3 
 5  2  3
3  6 3
Multiplying Radicals
To multiply radicals …multiply the inside by the inside
and the outside by the outside.
Then simplify.
6 3  4 2  (6  4)  ( 3  2)
 24 6
4 3  5 15  20 45
 60 5
Multiplying Radicals
It may be easier to simplify the radicals first.
5 80  6 32  20 5  24 2
16
5
16
 480 10
2
2 27  4 18  72 6
9
9
3
2
3 8  5 20
4
2
4
5
 60 10
Example 2
Multiply Radicals
Multiply the radicals. Then simplify if possible.
a. 3
·
7
b. 2
·
8
b. 2
·
8= 2· 8
SOLUTION
a. 3
·
7= 3· 7
= 21
= 16
=4
Squaring a Radical
Remember “squaring” and “square root” are inverse
operations.
 5 
 7 
 8 
 x 
2
5  5  25  5
2
7  7  49  7
2
8  8  64  8
2
x x 
x  x
2
Squaring a Radical

Evaluate the
expression
3 7 
3 7 

2
2
 32 
Evaluate the
expression
 2 11
2 2
2
 2 11  2  11
2
 7
2
9763
 4 11  44
Example 3
Find Side Lengths
Use the Pythagorean Theorem to find the
length of the hypotenuse to the nearest
tenth.
SOLUTION
a2 + b2 = c2
2
2 +
2
3 = c2
2 + 3 = c2
5 = c2
5=c
2.2 ≈ c
Write Pythagorean Theorem.
Substitute 2 for a and 3 for b.
Simplify.
Add.
Take the square root of each side.
Use a calculator.
Simplifying Division of Radicals
When dividing rational expressions with radical
components, and the denominator is a factor of the
numerator, consider the radicands and the coefficients
separately. Simplify the radicands and reduce the
coefficients.
4
20

5
2
42
27 6
3 2
9 3
10 22
5 2
11
 2 11
16 8
4
8 2
Simplifying Square Root
The properties of square roots allow us to
simplify radical expressions.
A radical expression is in simplest form
when:
1. The radicand has no perfect-square
factor other than 1
2. There’s no radical in the denominator
(rationalizing the denominator)
Step #1: The radicand has no perfect-square
factor other than 1.
Simplest Radical Form
Like the number
3/6, 75 is not
in its simplest
form. Also, the
process of
simplification for
both numbers
involves factors.

Method 1: Factoring
out a perfect square.
75 
25  3 
25  3 
5 3
Example Method 1 Simplifying Radicals
72  36  2
 36  2
6 2
1. Find the largest perfect
square that is a factor
of the radicand.
2. Rewrite the radicand as
a product of its largest
square and some other
number.
3. Take the square root of
the perfect square. Write
it as a product.
Perfect squares:
1, 4 , 9, 16, 25, 36,
49, 64, 81, 100,...
4. Leave the number that
you didn’t take the
square root of under
the radical sign.
Simplest Radical Form
In the second
method, pairs
of factors come
out of the
radical as
single factors,
but single
factors stay
within the
radical.

Method 2: Making a
factor tree.
75 
25 3
5 5
5 3
Simplest Radical Form
This method
works because
pairs of factors
are really
perfect squares.
So 5·5 is 52, the
square root of
which is 5.

Method 2: Making a
factor tree.
75 
25 3
5 5
5 3
Examples: Simplifying Radicals
50  5 2
25
2
150  5 6
25
6
288  12 2
144
2
121  11
11
11
Example 4
Simplify Radicals
Simplify the radical expression.
a. 12
b. 45
SOLUTION
a. 12 = 4 · 3
= 4 · 3
=2 3
b. 45 = 9 · 5
= 9 · 5
=3 5
Examples: Simplifying Radicals
6 27  6  27  6  3 3  18 3
9
3
4 200  40 2
100
2
5 80  20 5
16
5
3 64  24
8
8
Your Turn:
Express each square root in its simplest
form by factoring out a perfect square or
by using a factor tree.
12
18
24
32
40
2 3
3 2
2 6
4 2
2 10
48
60
75
83
4 3
2 15
5 3
83
300x3
10 x 3x
Your Turn:
Simplify the expression.
9
64
27
98
10  15
8  28
3 3
7 2
5 6
4 14
3
8
15
4
15
2
11
25
11
5
36
49
6
7
Step #2: There’s no radical in the denominator
(rationalizing the denominator).
Example 1
Evaluate, and then classify the product.
1.
(√5)(√5) = 5 (rational number)
2.
(2 + √5)(2 – √5) = 451 (rational number)
Conjugates
The radical expressions a + √b and a – √b
are called conjugates.

The product of two conjugates is always
a rational number
Example 2
Identify the conjugate of each of the
following radical expressions:
7
1. √7
2.
5 – √11 5 11
3.
√13 + 9 9 13
Rationalizing the Denominator
Recall that a radical expression is not in
simplest form if it has a radical in the
denominator. How could we use
conjugates to get rid of any radical in the
denominator and why?
Rationalizing the Denominator
We can use conjugates to get rid of radicals in
the denominator:
The process of multiplying the top and bottom of
a radical expression by the conjugate of the
denominator is called rationalizing the
denominator.
2
2 3 2 3
3



3
3
3 3
3
Fancy One
1 3

5 1 3

5  5 3 5  5 3




2
2
1 3 1 3 1 3 1 3
5
Fancy One



Example 3
Simplify the expression.
6
6
6 5
30



5
5
5
5 5
6
7 5

6 7  5

42  6 5


49 5
7 5 7 5




42  6 5 213 5

44
22
17
12
1
9 7

17
17 12


12
12 12
204 2 51 51


12
12
6



1 9  7

9  7 9  7 
9  7 9  7

81  7
74
Your Turn:
Simplify the expression.
9
8
2
4  11
3 2
4
82 11
5
19
21
399
21
4
32 4 3
61
8 3
Your Turn:
Find the square root. Round your answer to the nearest
tenth. Check that your answer is reasonable.
1. 27
ANSWER
2. 46
ANSWER 6.8; 36 < 46 < 49, so 6 < 6.8 < 7.
3. 8
ANSWER 2.8; 4 < 8 < 9, so 2 < 2.8 < 3.
4. 97
ANSWER 9.8; 81 < 97 < 100, so 9 < 9.8 < 10.
5.2; 25 < 27 < 36, so 5 < 5.2 < 6.
Your Turn:
Multiply the radicals. Then simplify if possible.
5. 3
·
6. 11
7. 3
·
·
8. 5 3
·
5
ANSWER
15
6
ANSWER
66
27
ANSWER
9
3
ANSWER
15
Your Turn:
Simplify the radical expression.
9.
20
ANSWER
2 5
10.
8
ANSWER
2 2
11. 75
ANSWER
5 3
12. 112
ANSWER
4 7
Assignment

Pg. 539 – 541; #1 – 61odd