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9.3 Simplifying Radicals
Square Roots
Opposite of squaring a number is taking the square root of a number.
A number b is a square root of a number a if b2 = a.
In order to find a square root of a, you need a # that, when squared, equals a.
In the expression , is the radical sign and
64 is the radicand.
If x2 = y then x is a square root of y.
1. Find the square root:
8 or -8
64
64
11, -11
4. Find the square root:
21 or -21
5. Find the square root:
3. Find the square root: 121
441
25
815
9
6.82, -6.82
6. Use a calculator to find each square root. Round the decimal answer to the nearest hundredth.
46.5
1 • 1 = 12 • 2 = 43 • 3 = 9
4 • 4 = 165 • 5 = 256 • 6 = 36
49, 64, 81, 100, 121, 144, ...
What numbers are perfect squares?
4
16
25
100
144
= 2
= 4
= 5
= 10
= 12
baab
0b if b
a
b
a
a bIf and are real numbers,
Product Rule for Radicals
Simplify the following radical expressions.
40 104 102
16
5 16
5
4
5
15 No perfect square factor, so the radical is already simplified.
Simplifying Radicals
Example
8
20
32
75
40
=
= =
=
=
€
4 • 2
€
4 • 5
€
16 • 2
€
25 • 3
€
4 • 10
=
=
=
=
=
22
52
24
35
102
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
48
80
50
125
450
=
= =
=
=
€
16 • 3
€
16 • 5
€
25 • 2
€
25 • 5
€
225 • 2
=
=
=
=
=
34
54
225
55
215
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
18
288
75
24
72
=
= =
=
=
=
=
=
=
=
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
1. Simplify
Find a perfect square that goes into 147. 147
147 349
147 349
147 7 3
2. Simplify
Find a perfect square that goes into 605.
605
121 5
121 5
11 5
Simplify
1. .
2. .
3. .
4. .
2 18
72
3 8
6 236 2
*To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals.
Multiply the radicals.
6. Simplify 6 10
60
4 154 152 15
7. Simplify 2 14 3 21Multiply the coefficients and radicals.
6 294
6 49 66 649
42 6
6 67
35*5 175 7*25 75
Multiply and then simplify
73*82 566 14*46
142*6 1412
204*52 10020 20010*20
€
5( )2
= 5*5 25 5
€
7( )2
= 7*7 49 7
€
8( )2
= 8*8 64 8
€
x( )2
= xx * 2x x
How do you know when a radical problem is done?
1. No radicals can be simplified.Example:
2. There are no fractions in the radical.Example:
3. There are no radicals in the denominator.Example:
8
1
4
1
5
To divide radicals: divide the coefficients, divide the radicands if possible, and rationalize the denominator so that no radical remains in the denominator
7
56 8 2*4 22
7
6This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator.
7
7*
7
6
49
42
7
42
42 cannot be simplified, so we are finished.
This can be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator.
10
5
2
2*
2
1
10
2
This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator.
12
3
3
3*
12
3
36
33
6
33
6
3Reduce the fraction.
8. Simplify.
Divide the radicals.
108
3
108
3
366
Uh oh…There is a
radical in the denominator!
Whew! It simplified!
9. Simplify
8 2
2 8
4
2
2
Uh oh…Another
radical in the denominator!
Whew! It simplified again! I hope they all are like this!
€
8 2
2 8•
8
8
€
8 16
2 • 8
10. Simplify
5
7
5
7
75
7 7
35
49 35
7
Since the fraction doesn’t reduce, split the radical up.
Uh oh…There is a fraction in the radical!
How do I get rid of the radical in
the denominator?
Multiply by the “fancy one” to make the denominator a
perfect square!
