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SKILL
Are You Ready?
53
Simplify Radical Expressions
Teaching Skill 53
Objective Simplify radical expressions.
Alternative Teaching Strategy
Objective Simplify radical expressions.
Review with students the definition of
simplest form. Ask: Is 4 written in
simplest form? (No) Why or why not? (4
1 written
is a perfect square factor.) Is __
7
in simplest form? (No, because there is a
5
fraction under the radical sign.) Is ___
3
written in simplest form? (Yes, even
Some students may benefit from seeing
the connection between square roots and
squares more directly.
Remind students that the first step in
simplifying a radical is to check for perfect
squares. If the number inside the radical is
the square of an integer, it can be simplified.
Write the following problem on the board:
2
49 7 7 7
though there is a fraction, the denominator
does not have a radical in it.)
Ask: Since taking the square root of a
number is the inverse of squaring the
number, what can be said about the
square root of a number squared? (It is
equal to the number.)
Next, review with students how to simplify
radical expressions. Work through each
example. Point out that when the expression
involves a product or a fraction, it may be
more convenient to multiply or divide first,
then simplify. Provide the following example:
20 5 . Ask: Is 20 or 5 a perfect square?
(No) If you multiply first, do you get a
perfect square inside the radical? (Yes,
100) Provide a similar example using a
45 ).
fraction (e.g. ___
5
Have students complete the exercises.
Have students complete the following table.
PRACTICE ON YOUR OWN
In exercises 1–8, students simplify radical
expressions.
CHECK
Determine that students know how to
simplify radical expressions.
n2
1
1
1
2
4
4
3
4
5
6
7
8
9
10
9
9
n 2 n
1 2 1
2 2 2
3 2 3
Write the problems below on the board.
2
Have students rewrite the problems as n
and then simplify. Remind students that if the
expression is a product, they can simplify
each term separately and then multiply.
Likewise, if the expression is a fraction,
they can simplify the numerator and the
denominator one at a time.
Students who successfully complete the
Practice on Your Own and Check are
ready to move on to the next skill.
COMMON ERRORS
Students may leave a radical expression in
the denominator of a fraction.
16
Students who made more than 2 errors in
the Practice on Your Own, or who were not
successful in the Check section, may benefit
from the Alternative Teaching Strategy.
Copyright © by Holt McDougal.
All rights reserved.
n
4 2 4 ; 81 9 2 9 ;
2
2
36 100 6 6, 10 10,
6 10 60 3 2 __3
9
9 ____
____
___
25 25
5 2 5
117
Holt McDougal Geometry
Name
Date
SKILL
Are You Ready?
53
Simplify Radical Expressions
Class
Definition: A radical expression is in simplest form when all of the following conditions are met.
1. The number, or expression, under the radical sign contains no perfect square
factors (other than 1).
2. The expression under the radical sign does not contain a fraction.
3. If the expression is a fraction, the denominator does not contain a radical expression.
How to Simplify Radical Expressions
Look for perfect square
factors and simplify these
first. If the radical expression
is preceded by a negative
sign, then the answer is
negative.
Example 1: Simplify 兹 81 .
Since 81 is a perfect
square factor, simplify the
expression to 9.
兹 81 兹 9 9 9
兹81 兹9 9 9
If the expression is a
product, simplify then
multiply, or multiply then
simplify, whichever is most
convenient.
Example 2: Simplify 兹25 兹 16 .
Since both numbers are
perfect squares, simplify then
multiply: 兹5 5 兹4 4 5 4 20
If the expression is (or
contains) a fraction, simplify
then divide, or divide then
simplify, whichever is most
convenient.
兹
4
Example 3: Simplify ___ .
49
兹2 2
4
2
__
___ ______
7
49
兹7 7
兹
Practice on Your Own
Simplify each expression.
1. 兹 25
2. 兹9 兹36
5. 兹100 兹 4
6. 兹2(32)
3.
兹121
81
____
4. 兹81
1
8. ____
625
7. 兹169
兹
Check
Simplify each expression.
9. 兹 16
10. 兹81 兹64
13. 兹2 兹 50
Copyright © by Holt McDougal.
All rights reserved.
14. 兹144
11. 兹 49
15. 兹 9 兹 4
118
12.
25
兹___
4
16.
36
兹___
9
Holt McDougal Geometry
SKILL
Are You Ready?
77
Solve Proportions
Teaching Skill 77
Objective Solve proportions.
Alternative Teaching Strategy
Objective Solve proportions.
Review with students the definition of a
proportion. Point out that you can also think
of a proportion as two equivalent fractions.
Tell students that it is possible in many
proportions to follow a pattern to solve the
proportion.
Ask: If two fractions are equivalent, what
is true about their simplest forms? (They
are equal.) Write two equivalent fractions on
5
1
the board, such as __ ___. Ask: Is this a
2
10
true statement and why? (Yes, because the
fractions are equivalent.) Tell students that
this is a proportion.
Write the following proportion on the board:
36
12 ___
___
. Ask: If you look at the
16
48
numerators, what are you multiplying by
to get from 12 to 36? (3) What are you
multiplying by in the denominators to get
from 16 to 48? (3)
Write the following on the board:
3
Show students by pointing what the cross
products of this proportion are. (1 10
and 2 5) Ask: What is true about the
cross products? (They are equal.) Explain
to students that this is the key to solving
proportions.
36
12 ___
___
16
3
Explain that because you are multiplying
both the numerator and the denominator by
the same number, you still have equivalent
3
ratios since __ 1.
3
2
Review with students the steps for solving a
proportion. Then work through the example.
Remind students that it does not matter
which side the variable is on when solving an
equation.
9
18
Write the following on the board: ___ ___
x
16
PRACTICE ON YOUR OWN
In exercises 1–12, students solve
proportions.
2
Have students find the value of x by
multiplying 16 times 2. (32)
CHECK
Determine that students know how to solve
proportions.
Tell students that this process also works
if you are dividing the numerator and
denominator by the same number. Write
8
72
the following on the board: ___ __
x . Have
45
students draw a diagram of the division, like
the multiplication diagrams above. ( 9 on
each piece; answer: x 5)
Students who successfully complete the
Practice on Your Own and Check are
ready to move on to the next skill.
COMMON ERRORS
Students may multiply the numerators
together and the denominators together,
rather than finding the cross products.
Have students use this technique to solve
8 ___
9
3
2
__
the following proportions: __ __
x ; 12 x ;
5
15 ___
30 11
18 __
6
88
77 ___
11
___
___
__
x ; __ ___
x ; 33 x ; and 48 x .
17
8
(x 20; x 4; x 34; x 56; x 11; x 6)
Students who made more than 3 errors in
the Practice on Your Own, or who were not
successful in the Check section, may benefit
from the Alternative Teaching Strategy.
Copyright © by Holt McDougal.
All rights reserved.
48
Then have students solve problems with x in
the numerator.
165
Holt McDougal Geometry
Name
Date
SKILL
Are You Ready?
77
Solve Proportions
Class
Definition: A proportion is an equation that shows two equivalent ratios.
Key property: The cross products of a proportion are equal.
To solve a proportion, follow these two steps:
• Step 1: Find the cross products.
• Step 2: Simplify if necessary and solve the equation for the variable.
4
x
Example: Solve __ ___
6
12
Step 1: Find the cross products.
Step 2: Simplify and solve.
4 ___
x
__
6
12
6 x 4 12
6 x 4 12
6x 48
Multiply.
Divide both sides by 6.
x8
Practice on Your Own
Solve each proportion.
2
x
1. __ ___
5
25
11
22
2. __ ___
x
4
8
x
3. ___ ____
16
160
100
x
4. ____ ____
500
100
9
45
5. ___ ___
x
10
90
x
6. ___ __
12
4
15
5
7. ___ __
x
7
30
x
8. ___ __
7
21
45
5
9. ___ __
x
8
1
x
10. ___ ___
99
33
8
2
11. __ __
x
9
8
32
12. __ ___
x
3
Check
Solve each proportion.
3
x
13. __ ___
7
49
15 5
14. ___ __
x
30
13
x
15. ___ __
2
4
66
6
16. ___ __
x
12
11
22
17. __ ___
x
4
6
x
18. __ __
5
4
1
x
19. ___ ___
55
44
12
24
20. ___ ___
x
5
Copyright © by Holt McDougal.
All rights reserved.
166
Holt McDougal Geometry
Name
Date
Class
Enrichment
CHAPTER
7
Angles and More Angles
A perfect number is a number which is the sum of its own positive factors (other
than itself). For example, the following numbers are perfect.
6⫽1⫹2⫹3
28 ⫽ 1 ⫹ 2 ⫹ 4 ⫹ 7 ⫹ 14
What is the next perfect number?
To discover the answer, find the value of x in each figure. Then, cross out the
answer in the box at the bottom of the page. The sum of all the remaining angles is
the next perfect number.
2.
1.
62°
x
3.
x
45°
5. x
37°
4.
x
x
x
6.
x
x
7.
8.
60°
x
3x
38°
x
2x
9.
3x
x
100°
10.
2x
x
11.
108°
110°
x
x
12.
x
115°
13.
14.
60°
5x
x
4x
13x
x
65⬚
45⬚
10⬚
20⬚
73⬚
90⬚
30⬚
83⬚
110⬚
60⬚
120⬚
53⬚
35⬚
71⬚
118⬚
115⬚
36⬚
40⬚
105⬚
72⬚
Copyright © by Holt McDougal.
All rights reserved.
201
Holt McDougal Geometry
Answer Key
continued
SKILL 53 ANSWERS:
11. 0
Practice on Your Own
12. 12
1. 5
Check
2. 18
9
3. __
11
4. ⫺9
13. 11
5. 20
16. 25
6. 8
17. 13
7. 13
18. 0
1
8. ⫺___
25
19. 1.1
Check
9. 4
10. 72
14. 2.3
15. 10
20. 1
SKILL 55 ANSWERS:
Practice on Your Own
11. ⫺7
2
12. __
5
13. 10
1. 3
2. 4
3. 31
14. ⫺12
4. 3
15. ⫺6
1
16. __
2
5. 6
6. 14
SKILL 54 ANSWERS:
7. 50
Practice on Your Own
8. 9
1. 15
9. 26
2. 8
10. 43
3. 0.4
11. 0
4. 1.19
12. 4
5. 10
Check
6. 4
13. 4
7. 0.75
14. 0
8. 0.7
15. 25
9. 6
16. 22
10. 7
17. 2
18. 58
Copyright © by Holt McDougal.
All rights reserved.
225
Holt McDougal Geometry
Answer Key
continued
SKILL 76 ANSWERS:
Check
Practice on Your Own
13. x ⫽ 21
1. perpendicular
14. x ⫽ 10
7. perpendicular
15. x ⫽ 26
12
16. x ⫽ ___
11
17. x ⫽ 8
15
18. x ⫽ ___
2
4
__
19. x ⫽
5
20. x ⫽ 10
8. perpendicular
SKILL 78 ANSWERS:
2. parallel
3. perpendicular
4. neither
5. parallel
6. neither
9. perpendicular
Practice on Your Own
Check
1.
10. perpendicular
11. parallel
12. parallel
13. parallel
14. neither
3.
15. perpendicular
SKILL 77 ANSWERS:
Practice on Your Own
1. x ⫽ 10
5.
2. x ⫽ 8
3. x ⫽ 80
4. x ⫽ 20
5. x ⫽ 50
6. x ⫽ 30
7
7. x ⫽ __
3
8. x ⫽ 10
8
9. x ⫽ __
9
1
__
10. x ⫽
3
11. x ⫽ 36
7.
x
y
⫺2
⫺1
0
1
2
⫺5
⫺2
1
4
7
x
y
⫺2
⫺1
0
1
2
4
1
0
1
4
x
y
⫺2
⫺1
0
1
2
0.5
1
1.5
2
2.5
x
y
⫺2
⫺1
0
1
2
0
1
4
9
16
2.
4.
6.
8.
x
y
⫺2
⫺1
0
1
2
13
9
5
1
⫺3
x
y
⫺2
⫺1
0
1
2
1
⫺2
⫺3
⫺2
1
x
y
⫺2
⫺1
0
1
2
5
4.5
4
3.5
3
x
y
⫺2
⫺1
0
1
2
16
9
4
1
0
12. x ⫽ 12
Copyright © by Holt McDougal.
All rights reserved.
233
Holt McDougal Geometry