Transcript Simplifying Radicals

Perfect Squares
1
64
225
4
81
256
9
16
100
121
289
25
36
49
144
169
196
400
324
4
=2
16
=4
25
=5
100
= 10
144
= 12
Simplify Using Perfect Square Method
147
Find a perfect square that goes into 147
147  49 3
147  49
147  7 3
3
Simplify
605
Find a perfect square that goes into 605
121 5
121
11 5
5
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
Simplifying Radicals by Tree Method.
Simplify the following.
20
Step one, break the number down into a factor tree.
20
10
2
Step two: Circle any pairs.
The pairs bust out of the klink.
2
5
So…
20
which is really just…
2 25
This guy gets away
This guy gets
caught and is
never heard from
again
so it ends up like this… 2
The guy that
got away
5
The guy left in
prison
This guy didn’t
have a partner
and so is left in
prison
But teacher,
teacher, what
happens to the
other guy?
He’s Dead, gone,
went bye bye,
won’t see him
again!
Another Example
Simplify the following:
140
Step 1: Make a factor tree
140
10
14
2
So that leaves:
Which is:
2 7 5
2 35
7
2
5
Cake Method
Video on Cake Method
Your Turn!
Simplify the following:
1.
44
2 11
2.
64
8
3.
56
2 14
4.
250
5 10
LEAVE IN RADICAL FORM
18
=
288 =
75
=
24
=
72
=
LEAVE IN RADICAL FORM
48
= 4
80
=
4 5
50
=
25 2
125
=
5 5
450
=
15 2
3
Adding or Subtracting Radicals
To add or subtract square roots you must have like radicands
(the number under the radical). Then you add or subtract the
coefficients!
2 3 2  4 2
Sometimes you must simplify first:
2  18  2  3 2  4 2
Example 1:
4 8  5 32 
4 4  2  5 16  2 
4 2 2  5 4 2 
8 2  20 2 
 12 2
Try These
3 5 5 5 
3 75  48 
4 5  5 18 
3 5 5 5 
2 20  3 80 
DO NOW
6 5  5 20 
18  7 32 
2 28  7  6 63 
*
To multiply radicals: multiply the
coefficients and then multiply
the radicands and then simplify
the remaining radicals.
Radical Product Property
a b
ab
ONLY when a≥0 and b≥0
For Example:
9  16  9 16  144  12
9  16  3  4  12
Equal
 5
2

5* 5 
25 

7* 7 
49  7

8* 8 
64  8

x* x 
x 
 7
2
 8
2
 x
2
2
5
x
Multiplying Radicals
1. Multiply terms outside the radical together.
2. Multiply terms inside the radical together.
3. Simplify.
12  5 3
 5 36
6 5  8  6 40
 6  2 10
56
 6 4 10
 12 10
 30
Multiplication and Radicals
Simplify the expression:
7 10  4 15
7  4  10  15
28 10 15
28 150
28 25 6
28  5 6
140 6
Multiply 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
Multiplying Radicals
You can multiply using distributive property and FOIL.
3 (7 
3)
 7 3 3
2 (5  8)  5 2  16
(6 
2 )(6 
5 2 4
2)  36 6 2  6 2  2
 36  2
 34
Multiply: You try.
5 (2  5 )
(2 
5)
2
 2 5 5
 ( 2  5 )( 2  5 )
4  2 5  2 5 5
 9 4 5
(5 
7 )(5  7)
 25  7
 18
Using the Conjugate to Simplify
Conjugate
Expression
Conjugate
(2  5 )
(2  5 )
(10 
(10 
2)
( 10  6 )
2)
( 10  6 )
Product
4  5  1
100  2  98
10  36   26
The radical “goes away” every time
To divide radicals:
divide the
coefficients, divide
the radicands if
possible, and
rationalize the
denominator so that
no radical remains in
the denominator
Radical Quotient Property
a

b
a
b
ONLY when a≥0 and b≥0
For Example:
64
16
64
16

64
16

 4 2
8
4
2
Equal
56

7
8
4* 2  2 2
Fractions and Radicals
Simplify the expressions:
a.
5 7
10
b.
There is nothing to
simplify because the
square root is
simplified and every
term in the fraction
can not be divided by
10.
Make sure to
simplify the
fraction.
4 12
2
4 4 3
2
4 2 3
2
2 2 3


2
2 3
c.
15 180
9
15 36 5
9
15 6 5
9
3 5 2 5


33
5 2 5
3
Dividing to Simplify Radicals
No radicals in the denominator allowed
Denominators must be “rationalized.”
Multiply by
2
3
15
5


3
3
5
5
√
1
in the form of
√
2 3

3
3
15 5
 1
5
3 5
Using the Conjugate to Simplify
Conjugate
Expression
Conjugate
(2  5 )
(2  5 )
(10 
(10 
2)
( 10  6 )
2)
( 10  6 )
Product
4  5  1
100  2  98
10  36   26
The radical “goes away” every time
Dividing to Simplify Radicals
conjugate
Multiply by
2
5 3
5
1
in the form of
5 3

5 3
1
10  2 3
 11
22
conjugate
10  2 3

25  3
5 3

11
Simplify: You try.
6
2
3 2
5
5
 5
4
2 3
84 3
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
Summary:
To ADD and SUBTRACT
To MULTIPLY
COMBINE LIKE TERMS
“Outside” NUMBERS x NUMBERS
“Inside” NUMBERS x NUMBERS
DISTRIBUTE and FOIL
To DIVIDE
“Rationalize” denominator using
Use conjugate
ALWAYS SIMPLIFY AT THE END IF YOU CAN
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