Download Simplifying Radicals

10-2
Simplifying Radicals
Vocabulary
Review
Find each square root.
1. !36 5 2. 4!144 5 100
36
3. 2Å 9 5 4.Å169 5 Vocabulary Builder
radical sign
radical (noun, adjective) rad ih kul
Definition: A radical is the indicated root of a quantity,
such as !7 or !x. The sign that indicates a root is to
be taken is a radical sign.
! 17
radical
radicand
Math Usage: A radical expression contains a radical. A radical expression is
simplified if the following are true:
HSM11_A1MC_1002_T91362
• The radicand has no perfect-square factors other than 1.
• The radicand contains no fractions.
• No radicals appear in the denominator of a fraction.
Use Your Vocabulary
Write T for true or F for false.
5. The expression !w 1 2 is a radical expression.
6. The radical expression !121 is in simplified form.
7. The radical expression 5!7 is in simplified form.
8. The radicand in the radical expression 5!7 is 5.
9. The radical expression
Chapter 10
5
is in simplified form.
!3
294 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Word Origin: from the Latin word “radix,” which means “root”
Property Multiplication Property of Square Roots
Algebra
Example
For a $ 0 and b $ 0, !ab 5 !a ? !b.
!48 5 !16 ? !3 5 4!3
10. Cross out the equation below that is NOT true.
!98 5 !49 ? !2
!200 5 !100 ? !2
!28 5 4!7 Problem 2 Removing Variable Factors
Got It? What is the simplified form of 2m"80m9?
11. Circle the factor tree that shows the greatest perfect square factor of 80.
80
80
20
4
40
80
2
16
80
5
8
12. Circle the expression that should be used to simplify 2m"80m9 .
2m"40m8 ? "2m
2m"16m8 ? "5m
10
2m"16m9 ? "5 HSM11_A1MC_1002_T91364
HSM11_A1MC_1002_T91363
HSM11_A1MC_1002_T91365
HSM11_A1MC_1002_T91366
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
3. Use your expression from Exercise 12 to simplify 2m"80m9 .
1
Problem 3 Multiplying Two Radical Expressions
Got It? What is the simplified form of the expression 3!6 ? !18?
14. Complete the reasoning model below.
Write
Think
I need to multiply the radicals first.
3œ6 ∙ œ18 = 3 ∙
To simplify the radical, I need to think of the
greatest perfect square factor of 108. Then I will
3∙
√
use the Multiplication Property of Square Roots.
√
= 3 ∙√
= 3∙
Finally, I multiply the coefficients of the radicals.
=
HSM11_A1MC_1002_ T91367
∙
= 3∙
Now I simplify the radicals.
295
√
∙
∙
√
√
√
∙
Lesson 10-2
Problem 4 Writing a Radical Expression
Got It? A door’s height is four times its width w. What is the maximum length of a
painting that fits through the door?
15. When moving something tall through a doorway, how would you position the
object to get the maximum amount of room for that object? Explain.
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16. Based on your answer to Exercise 15, circle the measurement that you need to
determine to fit a painting through a doorway.
width of doorway
height of doorway
diagonal length of doorway 17. Let w represent the width of the door. Circle the expression that represents the
height of the door.
w
w 1 4
4w
4w 1 4 18. Use the Pythagorean Theorem to find the length of the diagonal of the doorway.
doorway times . The maximum length of a painting that fits through
the door is w ? , or about .
Property Division Property of Square Roots
Algebra
Example
a
!a
.
For a $ 0 and b . 0, Åb 5
!b
36
Å49
5
!36
6
57
!49
Problem 5 Simplifying Fractions Within Radicals
Got It? What is the simplified form of the radical expression Å144
9 ?
20. Write the justifications to simplify the radical expression.
144
Å 9
5
!144
!9
5 12
3
5 4
Chapter 10
296 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
19. The length of the diagonal of the doorway is equal to the width of the
Problem 6 Rationalizing Denominators
Got It? What is the simplified form of the radical expression
21. Complete the steps to simplify the expression.
!2
!2
5
?
!3
!3
5
Å
Å
!2
?
!3
Multiply the numerator and denominator by the radical
you want to eliminate.
Å
Simplify your result.
Simplify completely.
Å
5
Å
Lesson Check • Do you UNDERSTAND?
Compare and Contrast Simplify
3
two different ways. Which way do you prefer? Explain.
!12
22. Simplify the expression. Then rationalize. 23. Rationalize the denominator. Then simplify.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
24. Which way do you prefer to simplify rational expressions? Explain.
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Math Success
Check off the vocabulary words that you understand.
radical expression
radical
radicand
rationalize the denominator
Rate how well you can simplify radicals.
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review
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get it!
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Lesson 10-2