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Simplifying Radicals
Section 10-2 Part 1
Goals
Goal
• To simplify radicals
involving products.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• Radical Expression
Definitions
• Radical Expression - an expression that contains a
radical sign
.
– There are many different types of radical expressions,
but in this course, you will only study radical
expressions that contain square roots.
– Examples:
• The expression under a radical sign is the
radicand. A radicand may contain numbers,
variables, or both. It may contain one term or
more than one term.
Simplest Form
Simplest Form
Simplified
3 5
9 x
2
4
Not Simplified
3 12
Radicand contains a
perfect square factor.
x
2
Radicand contains
a fraction.
5
7
Radical appears in the
denominator of a fraction.
Multiplying Radicals
Example: Multiplying
Radicals
1.
2 5  25  10
2. 7  7  77  49 7
3.
 x
2
 x  x  x x  x 2  x
Your Turn:
1.
3  7  37  21
  121 11
2. 11 11  1111
 3
2
3.
 3  3  33  9  3
The radicand has no perfect-square factors other than 1.
SIMPLIFYING RADICALS
REMOVING PERFECT-SQUARE FACTORS
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.
Removing Perfect-Square
Factors
Like the number
3/6, 75 is not in
its simplest form.
Also, the process
of simplification
for both numbers
involves factors.
• Factoring out a perfect
square.
75 
25  3 
25  3 
5 3
Procedure: Removing Perfect-Square Factors
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.
Example:
Simplify.
Factor the radicand using perfect squares.
Product Property of Square Roots.
Simplify.
Examples:
50  5 2
25
150  5 6
25
6
2
288  12 2
144
2
Your Turn:
Simplify.
Factor the radicand using perfect squares.
Product Property of Square Roots.
Simplify.
Your Turn:
Simplify.
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
300
10 3
Removing Variable
Factors
Example: Simplify the following:
x15  x14 x
 x7 x
To simplify a variable with an exponent, write the
product of the form
xn x where n is the
largest possible even exponent.
As a general rule, divide the exponent by two.
15 ÷ 2= 7 with remainder 1. The remainder
stays in the radical.
Example: Removing
Variable Factors
Simplify.
Product Property of Square Roots.
Product Property of Square Roots.
Example: Removing
Variable Factors
Simplify.
Factor the radicand using perfect squares.
Product Property of Square Roots.
Simplify.
Your Turn:
Simplify.
Product Property of Square Roots.
Product Property of Square Roots.
12.1 and 12.2 Operations with Radicals
Your Turn:
√ 225x2
= 15x
√ 25y6
= 5y3
√ 98m3 = 7m√2m
√ 27x4 = 3x2√3
2
√ 72x2y5 = 6xy √2y
Multiplying Radicals Expressions
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
 20 9  5
 60 5
Your Turn:
It may be easier to simplify the radicals first.
5 80  6 32
16
5
16
 20 5  24 2
 480 10
2
2 27  4 18
9
9
3
2
3 8  5 20
4
2
4
 72 6
5
 60 10
Example: Multiply Radical
Expressions
Multiply and simplify.
2 5x 11 2x  22 10x 2  22x 10
Multiply the
coefficients.
Multiply the
radicals.
Example:
Multiply and simplify.
3 2 x 4 50 x 
2
12 100x 
1210x
120x
Your Turn:
Multiply and simplify.
3
3
4
6 2 x 2 8 x y 
6
4
12 16x y 
3 2
3
2
124x y  48x y
Your Turn:
Multiply and simplify.
2
2
3
2 10 xy 3 5 x y 
3
5
6 50x y 
2
4
6 225 x x  y  y 
2
2
65 xy 2 xy  30 xy 2 xy
Multiply
Multiply and simplify.
1.
3
8 5 x 3 20 x
2
240x
3
2
2
6
2. 5 12 x y 3 3 x y
2
4
90x y x
Assignment
• 10-2 Pt 1 Exercises Pg. 613: #6 – 30 even