Chapter 11
Radical Expressions
SPI 3102.2.1- Operate (add, subtract, multiply, divide, simplify) with radicals and radical expressions
including radicands involving rational numbers and algebraic expressions.
Section 11-6: Simplifying Radical Expressions
Objective: To simplify radical (square root) expressions.
You can use radical expressions to find the length of a throw in baseball.
Radical Expression: An expression that contains a _________________________.
Radical Symbol
Index
3
√125
Radicand
How to break down a radical expression:
Example 1: Simplify √32.
Step 1: Find the prime factorization of the
numbers (and variables) in the expression.
√32
4
8
2 2 2 4
2 2
Step 2: Group all like factors and variables
in sets equal to the index. (In this example, the
index is 2, so group them in 2’s)
√2 ∙ 2 ∙ 2 ∙ 2 ∙ 2
√22 ∙ 22 ∙ 2
Step 3: Break up your radical over the multiplication.
√22 ∙ √22 ∙ √2
Step 4: Cancel any operations.
√22 ∙ √22 ∙ √2
2 ∙ 2 ∙ √2 = 4√2
3
Example 2: Simplify √48.
You Try!
3
2. Simplfy √216
1. Simplify √45
How to know if a radical expression is completely simplified;



The radicand has no perfect factors other than 1.
The radicand has no fractions. (No fractions under the
fraction.)
There are no square roots in the denominator.
, but your answer CAN be a
How to simplify radicals when there are variables:
Example 1: Simplify √48𝑥 2 𝑦 3 .
√48𝑥 2 𝑦 3
Step 1: Find the prime factorization of the
numbers (and variables) in the expression.
6
8 𝑥∙𝑥 𝑦∙𝑦∙𝑦
2 3 2 4
2 2
Step 2: Group all like factors and variables
in sets equal to the index. (In this example, the
index is 2, so group them in 2’s)
√2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦
√22 ∙ 22 ∙ 3 ∙ 𝑥 2 ∙ 𝑦 2 ∙ 𝑦
Step 3: Break up your radical over the multiplication.
√22 ∙ √22 ∙ √3 ∙ √𝑥 2 ∙ √𝑦 2 ∙ √𝑦
Step 4: Cancel any operations.
√22 ∙ √22 ∙ √3 ∙ √𝑥 2 ∙ √𝑦 2 ∙ √𝑦
2 ∙ 2 ∙ √3 ∙ 𝑥 ∙ 𝑦 ∙ √𝑦 = 4𝑥𝑦√3𝑦
3
Example 2: Simplify √54𝑎3 𝑏 5 .
You Try!: Simplify √50𝑥 3 .
What about fractions?
Example 1: Simplify √
3𝑏2
27𝑏4
1
Step 1: Simplify the fraction.
√
Step 2: Take the square root of the numerator
√1
√9𝑏2
9𝑏2
There is nothing left
under the radical
because everything
cancels out!
and denominator separately.
Step 3: Break it down and simplify like we
√1
√32 𝑏2
=
1
3𝑏
did on the previous examples.
Example 2: Simplify √
9𝑦 6
36𝑦 2
Example 3: Simplify √
YOU TRY!!
1. Simplify √
40𝑚3
10𝑛4
2. Simplify √
128
EOC PREP:
1) Write 18 x 4 y 5 in simplest radical form.
A. 2x2y2 3 y
C. 2xy 3y 2
B. 3x2y2 6 y
D. 3x2y2 2 y
25
250𝑞10
5𝑞4
Section 11-7: Adding and Subtracting Radicals
Objective: To add and subtract radical expressions, which can be used to find the perimeter of
a figure.
Like radicals: Square root expressions with the same radicand.
Tell whether the following are like radicals:
1)
2 5and4 5 ____________
2) 12 2and12 5 ______________
You can only add or subtract LIKE radicals!
Example 1: 4√𝑦 + 6√𝑦 = ________
Example 3: √108 + √75
54
2 3 25
6 9
Example 2: 6√15 − √15 = ________
Sometimes they won’t be like terms but you will have to
break down the radicals if you can and then see if you can
combine them.
5 5
2 3 3 3
√22 ∙ 32 ∙ 3 + √3 ∙ 52
2 ∙ 3 ∙ √3 + 5 ∙ √3 = 6√3 + 5√3 = ________
YOU TRY!!!
1. √45 + √180
Challenge: 4√52𝑥 + √117𝑥 − 2√13
2. −2√3𝑏 + √27𝑏
Section 11-8(A): Multiplying Radicals
Objective: To multiply radical expressions.
Multiplying 2 radicals together:
Example 1: √12 ∙ √5
Step 1: Multiply the numbers under the radical together.
√12 ∙ 5 = √60
Step 2: Break down the radicand and simplify.
√60
√2 ∙ 2 ∙ 3 ∙ 5 = √22 ∙ 3 ∙ 5
2√3 ∙ 5 = 2√15
Example 2: (3√6)
2
YOU TRY!!
1. (2√7)
2
2. 4√7𝑥 ∙ √20𝑥
Section 11-8(B): Rationalizing Radicals
Objective: To learn what to do if there is a square root in the denominator.
If there is a square root in the denominator of a fraction then the radical expression is not
simplified. To get rid of the root in the denominator we have to do something called rationalize
the denominator.
Example 1:
√5
√6
Step 1: Multiply the numerator and denominator
√5 √6
∙
√6 √6
=
√5∙6
√6∙6
√30
√62
√30
6
by the radical on the bottom.
Step 2: Simplify the radicals on top and bottom.
=
=
√30
√36
Example 2:
√10
√11𝑥
You Try!
√7
1.
√15
2.
Challenge: −
√32
√48𝑧
√75𝑘
10√2𝑘
EOC PREP:
1) Which expression is equivalent to
10x
5
?
2) What is the product of 2 3 and 3 5 ?
A. 2x 5
B. 10x 5
A. 5 15
B. 6 8
C. 5x 2
10x
D.
25
C. 5 8
3) Write
A. 5
B. 15
C. 3
D. 5
75 in simplest radical form.
3
5
5
15
D. 6 15