Ch 9 Algebra Review L1 Key
Algebra Review for Chapter 9: Radicals
2
3  3  3  3 or
3 3  9 3
 9
Definition: square root is the inverse of squaring.
If x is a number greater than or equal to 0,
x
x * x  x and x 2 and ( x ) 2  x
9
42  4
Operations:
Multiplying Radicals
3 * 5  15
Dividing Radicals:
15
Simplest Radical Form:
Factor out perfect squares from under the radical sign.
3
Memorize
Perfect
Squares:
Radicals aka Square Roots:
Examples:
2
Name ___________________________

15
 5
3
a * b  ab “You may multiply radicals.”
a
b

a
or
b
a  b  a  b “You may divide radicals.”
Write answers in “simplest radical form”:
Factor into perfect squares & remove perfect squares from under the radical.
Example 1.
Example 2.
Simplify
Simplify
50
break into factors
25  2
25  2 to find perfect squares
take the square root
5 2
write: 5 2 multiplication implied
48
break into factors
4 12
4  4  3 keep going
16  3 find perfect squares
take the square root
4 3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
x2
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
A. Simplify by taking factors that are perfect squares out from under the radical. You should NOT need a
calculator!
1.
2.
6.
49 = 7
196 = 14
80 = 4 5
3.
20 = 2 5
4.
5.
32 = 4 2
98 = 7 2
 45
 442
 49  2
 4 5
 16  2
7 2
2 5
4 2
7.
700 = 10 7
8.
9.
75 = 5 3
270 = 3 30
 8  10
 100  7
 553
 27  10
 4225
 100  7
 25  3
 9 35 2
 16  5
 10 7
5 3
 9  30
 3 30
4 5
When you calculate with radicals, you should ALWAYS simplify your answers!!
S. Stirling Spring 2016
Page 1
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Ch 9 Algebra Review L1 Key
Name ___________________________
B. Use the formula a2  b2  c2 and then solve the resulting
equation. Write answers in simplest radical form.
1. If b  5 and c  8 , find a.
2. If a  3 and b  4 , find c.
a 2  b2  c2
a 2  b2  c2
a 2  52  82
32  42  c 2
a 2  25  64
9  16  c 2
a 2  39
25  c 2
a  39
Example
If a  3 and c  7 , find b.
a 2  b2  c 2
32  b2  72
9  b2  49
9  b2  49
b2  40
substitute values
simplify
subtract 9 both sides
take the square root
both sides
now
simplify
b  40
40  4 10  2 10
c  25  5
3. If b  2 and c  8 , find a.
4. If a  2 and c  9 , find b.
5. If a  4 and b  6 , find c.
a 2  b2  c2
a 2  b2  c2
a 2  b2  c2
a 2  2 2  82
22  b 2  92
42  62  c 2
a 2  4  64
4  b 2  81
16  36  c 2
a 2  60
a  60  4 15  2 15
52  c 2
c  2  2 13  2 13
b 2  77
b  77
C. Compute (or evaluate) the expressions. Simplify your answers. Examples:
You may multiply & divide radicals!
1.
4 8
2.
3.
3 12
8 * 18
1.
442
2429
33 4
449
94
33  6
4 2
16  9
4 15 find perfect squares
2 15 take square root
4  3  12
4. 2 2  3 18 
39
5.
2 3 2  2 9
2 3 2 3
39
4  9  36
3
7.
2 18
3 2
=
2 9 23

2
3
3
S. Stirling Spring 2016
45
6.
3
 13
5
8.
5 14
10 7
=
2. 2 2  5 6
5
45
2
2
6  10
Multiply numbers under the
radical, leave in factored form.
2  3  2  5 factored
2  5  2  6 organize factors
10  2 2 3 multiply &
 9 3
10  2  3 find perfect squares
simplify
20 3
3.
6 15
3 5
2 3
since 5 divides into 15
& 3 divides into 6…
Page 2
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Ch 9 Algebra Review L1 Key
Name ___________________________
D. Use the formula h  l 2 and then solve the resulting
equation. Write answers in simplest radical form.
1. If h  3 32 , find l.
hl 2
2. If l  3 5 , find h.
hl 2
3 32  l 2
h3 5 2
3 16  l
l  3  4  12
h  3 10
3. If h  30 , find l.
4. If l  2 10 , find h.
hl 2
hl 2
h  2 10  2
30  l 2
h  2 20
15  l
h  2 45
h4 5
Use h  l 2 .
Examples
1. If h  5 2 , find l.
hl 2
5 2  l 2 substitute values
5 2 l 2
divide both sides by

2
2
simplify
5l
2
2. If l  5 6 , find h.
hl 2
h5 6 2
substitute values
simplify completely
h  5 12
h  5 4  3  5  2 3  10 3
E. Use the formulas h  2s and l  s 3 and then solve the
resulting equations. Write answers in simplest radical form.
1. If l  2 15 , find h and s.
ls 3
2. If s  3 6 , find h and l.
Examples Use h  2s and l  s 3 .
h  2s
If h  4 6 , find l and s.
h  23 6
2 15  s 3
h6 6
2 5s
h  2s
l 3 6 3
h  22 5
l  3 2 33
h4 5
l  3 3 2  9 2
3. If h  6 , find l and s.
4. If l  5 6 , find h and s.
h  2s
6  2s
s
6
2
ls 3
6
l
 3
2
18 3 2
l

2
2
ls 3
5 6s 3
5 2s
h  2s
4 6  2s
start with h = formula
substitute values
4 6 2s
divide both sides by 2

2
2
simplify
2 6s
now you have s, so
ls 3
l  2 6  3 substitute values
simplify
l  2 18
l  2 92
l  2  3 2  6 2
h  2s
h  25 2
h  10 2
S. Stirling Spring 2016
Page 3
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Ch 9 Algebra Review L1 Key
Name ___________________________
Practice with Radicals
Directions: Use the properties given above to operate on and to completely simplify the radical expressions
1.
49 = 7
2.
196 = 14
3.
20 = 2 5
4.
5.
4 8
6.
8 * 18
7.
3 12
8. 2 2  3 18 
4 4 2 4 2
64
16
9.
64
8
 2
16 4
13.
56
8
7
4 2 2 9  4 3  12
96
2
10.
48  4 4 3  4 3
7
16
14.
7
16

7
4
17. Think! x 2 means x  x so how could you find
2
 2 3  ? Write 2 3  2 3 then simplify!
33 4 3 26
11.
50
8
18
2
18
 9 3
2
18.
5 3 
2 3 2 2 9  6 6  36
96
12.
25
25 5


4
2
4
15.
32 = 4 2
2
8
12  4 3  2 3
16.
4 18

3
2
18 4
6 2
3 2
19.  2 5 
2 32 3
5 3 5 3
2 52 5
4 9
4  3  12
25 9
25  3  75
4 25
4  5  20
21.  3  2    4  2 
22.
3 2  4  2
1  2 2
2 3
20. Think! You can only add (or subtract) like
things. So if 3x  4x  7 x , what is
3 54 5 ?
6 2 3 6
7 5
S. Stirling Spring 2016
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