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Simplifying, Multiplying, & Rationalizing Radicals
Homework: Radical Worksheet
Perfect Squares
1
4
916
253649
64
81
100121
144169196
225
256
324
400
625
289
…
4
16
25
100
144
= 2
= 4
= 5
= 10
= 12
8
20
32
75
40
=
= =
=
=
2*4
5*4
2*16
3*25
10*4
=
=
=
=
=
22
52
24
35
102
Perfect Square Factor * Other FactorL
EA
VE
IN
RA
DIC
AL
FO
RM
48
80
50
125
450
=
= =
=
=
3*16
5*16
2*25
5*25
2*225
=
=
=
=
=
34
54
25
55
215
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
Simplify32
216
4
24
Simplify 45
59
3
53
Simplify 96
16 6
64
4
2 3 6
Simplify
216
4 54
66
69OR
6
216
36 6
66
+To combine radicals: ADD the coefficients of like radicals
Simplify each expression
737576 78
62747365 7763
Simplify each expression: Simplify each radical first and then combine.
323502 2*1632*252
22
212210
24*325*2
Now you have like terms to combine
Not like terms, they can’t be combined
Simplify each expression: Simplify each radical first and then combine.
485273 3*1653*93
329
32039
34*533*3
Now you have like terms to combine
Not like terms, they can’t be combined
*To multiply radicals:
1. multiply the coefficients 2. multiply the radicands 3. simplify the remaining
radicals.
35*5 175 7*25 75
Multiply and then simplify
73*82 566 14*46
142*6 1412
204*52 1008 8010*8
2
5 5*5 25 5
2
7 7*7 49 7
2
8 8*8 64 8
2
x xx * 2x x
Short cut
Squaring a Square Root
Short cut
2
6222 62 2464
2
53 22 53 4559
2
5
3
2
2
5
3
25
3
Squaring a Square Root
To divide radicals:
-divide the coefficients
-divide the radicands, if possible
-rationalize the denominator so that no radical remains in the
denominator
63
26
There is an agreement
31
in mathematics that we don’t leave a radical
in the denominator of a fraction.
So how do we change the radical denominator of a fraction?
31
(Without changing the value of the fraction) The same way we change the denominator of any fraction…
41
12
3
3
3
4
1
For Example:
Multiply by a form of 1.
By what number can we multiply
to change to a rational number? 3
13
The answer is . . . . . . by itself!
3
1
3 3 23 3
3
3
33
31
3
3
Squaring a Square Root gives the Root!
3
133
Because we are changing the denominator
we call this process rationalizing.
to a rational number,
2
4
2224
Rationalize the denominator:
2
24
(Don’t forget to sim
plify)22
2
2
2
2
4
12
64
128
1212128
Rationalize the denominator:
36
3
1296
(Don’t forget to sim
plify)
(Don’t forget to sim
plify)
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
7
568
2*4 22
Simplify.7
56
Divide the radicals.
Simplify.
Simplify.
Divide the radicals.
108
3
108
3
366
Uh oh…There is a
radical in the denominator!
Whew! It simplified!
Simplify
8 2
2 8
4 1
4
4
2
2
Uh oh…Another
radical in the denominator!
Whew! It simplified again! I hope they all are like this!
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 1” to make the denominator a
perfect square!
*
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.
Fractional form of “1”
10
5
2
2*
2
1
2
2
Simplify fraction
Rationalize Denominator
2
1
12
3
3
3*
12
3
36
33
6
33
2
3Reduce the
fraction.
Use any fractional form of “1” that will result in a perfect
square
Finding square roots of decimals
If a number can be made be dividing two square numbers then we can find its square root.
For example,
= 3 ÷ 10
= 0.3
0.09 = 9 ÷ 100
Find 0.09
= 12 ÷ 10
= 1.2
1.44 = 144 ÷ 100
Find 1.44
If a number cannot be written as a product or quotient of two square numbers then its square root cannot be found exactly.
Use the key on your calculator to find out 2.
The calculator shows this as 1.414213562
This is an approximation to 9 decimal places.
The number of digits after the decimal point is infinite.
Approximate square roots
Estimating square roots
What is 10?
10 lies between 9 and 16.
Therefore,
9 < 10 < 16
So,
3 < 10 < 4
Use the key on you calculator to work out the answer.
10 = 3.16 (to 2 decimal places.)
10 is closer to 9 than to 16, so 10 will be
about 3.2
Suppose our calculator does not have a key.
36 < 40 < 49
So,
6 < 40 < 7
6.32 = 39.69 too small!
6.42 = 40.96 too big!
Trial and improvement
Find 40 40 is closer to 36 than to 49, so 40 will be about 6.3
6.332 = 40.0689 too big!
6.322 = 39.9424 too small!
Suppose we want the answer to 2 decimal places.
6.3252 = 40.005625 too big!
Therefore,
6.32 < 40 < 6.325
40 = 6.32 (to 2 decimal places)
Trial and improvement