Transcript Simplify Radicals - Nutley Public Schools

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.
If x2 = y then x is a square root of y.
In the expression 64 ,
is the radical sign and
64 is the radicand.
1. Find the square root: 64
8 or -8
3. Find the square root:  121
11, -11
4. Find the square root:
21 or -21
5. Find the square root:
5

9
441
25

81
6. Use a calculator to find each
square root. Round the decimal
answer to the nearest hundredth.
 46.5
6.82, -6.82
What numbers are perfect squares?
1•1=1
2•2=4
3•3=9
4 • 4 = 16
5 • 5 = 25
6 • 6 = 36
49, 64, 81, 100, 121, 144, ...
4
=2
16
=4
25
=5
100
= 10
144
= 12
Product Rule for Radicals
If
a and b are real numbers,
ab  a  b
a
a

if
b
b
b 0
Simplifying Radicals
Example
Simplify the following radical expressions.
40 
4  10  2 10
5

16
5
5

4
16
15
No perfect square factor, so the
radical is already simplified.
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor



8
=
4 
2=
2 2
20
=
4 
5=
2 5
32
=
16 
2=
4 2
75
= 25 
3=
5 3
40
= 4 
10= 2 10
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor



48
= 16 
3=
4 3
80
= 16 
5=
4 5
50
=
125
= 25 
450
= 225 
25 
2= 25 2
5=
5 5
2= 15 2
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
18
=
=
288
=
=
75
=
=
24
=
=
72
=
=
1. Simplify
147
Find a perfect square that goes into 147.
147  49 3
147  49
147  7 3
3
2. Simplify 605
Find a perfect square that goes into 605.
121 5
121
11 5
5
Simplify
1.
2.
3.
4.
2 18
.
3 8
6 2
36 2
.
.
.
72
*
To multiply radicals: multiply the
coefficients and then multiply the
radicands and then simplify the
remaining radicals.
6. Simplify 6  10
Multiply the radicals.
60
4 15
4
15
2 15
7. Simplify 2 14  3 21
Multiply the coefficients and radicals.
6 294
6 49 6
6
49
67
6
6
42 6
Multiply and then simplify
5 * 35  175  25 * 7  5 7
2 8 * 3 7  6 56  6 4 *14 
6 * 2 14  12 14
2 5 * 4 20  20 100  20 *10  200
 5 
5* 5 
25 
 7 
7* 7 
49  7
 8 
8* 8 
64  8
2
2
2
 
2
x
 x* x 
x 
2
5
x
How do you know when a radical
problem is done?
1. No radicals can be simplified.
Example:
8
2. There are no fractions in the radical.
1
Example:
4
3. There are no radicals in the denominator.
Example:
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
56

7
8
4* 2  2 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.
6

7
6
*
7
42

49
7

7
42
7
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.
5

10
1
*
2
2
10
2

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.
3

12
3
*
12
3

3
3 3

36
Reduce
the
fraction.
3 3

6
3
6
8. Simplify.
Whew! It
simplified!
108
3
Divide the radicals.
108
3
36
6
Uh oh…
There is a
radical in the
denominator!
8
2
9. Simplify
2 8
8 2
8

2 8
8
Whew! It simplified
again! I hope they
all are like this!
8 16
28


4
2
2
Uh oh…
Another
radical in the
denominator!
10. Simplify
5
7
Uh oh…
There is a
fraction in
the radical!
Since the fraction doesn’t reduce, split the radical up.
5
7
5

7
How do I get rid
of the radical in
the denominator?
7
7
35

49
Multiply by the “fancy one”
to make the denominator a
perfect square!
35

7