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Simplifying, Multiplying, &
Rationalizing Radicals
Homework:
Radical
Worksheet
Perfect Squares
1
64
225
4
81
256
9
16
100
121
289
25
36
49
144
169
196
400
324
…
625
4
=2
16
=4
25
=5
100
= 10
144
= 12
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
=
25 * 2 =
5 2
125
=
25 * 5 =
5 5
450
=
225 * 2 = 15 2
Simplify
32 
16 
4
4 2
2
45 
Simplify
9 
3
3 5
5
96 
Simplify
16

4
4 6
6
Simplify
216
216
54
4
6
9
6
36
OR
6
6 6
2  3 6
6 6
+
To combine radicals: ADD the
coefficients of like radicals
Simplify each expression
6 7 5 7 3 7 
8 7
5 6 3 7 4 7 2 6 
3 6 7 7
Simplify each expression:
Simplify each radical first and then combine.
2 50  3 32  2 25 * 2  3 16 * 2 
Not like terms, they
can’t be combined
Now you have like
terms to combine
2 * 5 2  3* 4 2 
10 2 12 2 
2 2
Simplify each expression: Simplify each
radical first and then combine.
3 27  5 48  3 9 * 3  5 16 * 3 
Not like terms, they
can’t be combined
Now you have like
terms to combine
3* 3 3  5 * 4 3 
9 3  20 3 
29 3
*
To multiply radicals:
1. multiply the coefficients
2. multiply the radicands
3. simplify the remaining radicals.
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  8 100 
8 *10  80
Squaring a Square Root
 5
2

5* 5 
25  5

7* 7 
49  7

8* 8 
64  8

x* x 
x 
 7
2
 8
2
 x
2
2
x
Short cut
Short cut
Squaring a Square Root
2 6 
2
2  6
2
 3 5 
2
2
4  6  24
  3  5  9  5  45
2
2
2
 3

 
 5 


2
3

2
5
3
25
To divide radicals:
-divide the coefficients
-divide the radicands, if
possible
-rationalize the
denominator so that no
radical remains in the
denominator
3
6
6
2
There is an agreement
in mathematics
that we don’t leave a radical
in the denominator
of a fraction.
1
3
So how do we change the radical denominator
of a fraction?
(Without changing the value of the fraction)
The same way we change the denominator of
any fraction… Multiply by a form of 1.
For Example:
1
3
1 1 3 3
  
4 4 3 12
By what number can we multiply
to change
1
3
to a rational number?
The answer is . . . . . . by itself!
3  3

2
3 
3
Squaring a Square Root gives the Root!
1
3
1
3
 

3 3
3 3
3
3
3
Because we are changing
the denominator
to a rational number,
we call this process
rationalizing.
3
1

3 3
Rationalize the denominator:
4

2
4
2


2
2
4 2

2 2
24 2
2 2
2
Rationalize the denominator:
8
96
8 12



12
12 12 12
6
4 6

12
3
3
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
Simplify.
56
7
Divide the radicals.
56

7
Simplify.
4*2 
2 2
8
Simplify.
Whew! It
simplified!
108
3
Divide the radicals.
108
3
36
6
Uh oh…
There is a
radical in the
denominator!
Simplify
8 2
2 8
4 1
4
Whew! It simplified
again! I hope they
all are like this!
4
2
2
Uh oh…
Another
radical in the
denominator!
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

*
7 7
How do I get rid
of the radical in
the denominator?
35

49
Multiply by the “fancy 1” to
make the denominator a
perfect square!
35

7
Fractional form of “1”
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.
5

10
Simplify fraction
Rationalize
Denominator
1
*
2
2

2
1
2
2
2
3

12
3
*
12
3 3

36
Reduce
the
fraction.
3

3
Use any fractional
form of “1” that will
result in a perfect
square
3 3

6
3
2
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,
Find 0.09
0.09 = 9 ÷ 100
Find 1.44
1.44 = 144 ÷ 100
= 3 ÷ 10
= 12 ÷ 10
= 0.3
= 1.2
Approximate square roots
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.
Estimating square roots
What is 10?
10 lies between 9 and 16.
Therefore,
9 < 10 < 16
10 is closer to 9
than to 16, so 10
will be about 3.2
So,
3 < 10 < 4
Use the 
key on you calculator to work out the answer.
10 = 3.16 (to 2 decimal places.)
Trial and improvement
Suppose our calculator does not have a
Find 40
36 < 40 < 49
So,
6 < 40 < 7
6.32 = 39.69
too small!
6.42 = 40.96
too big!

key.
40 is closer to 36
than to 49, so 40
will be about 6.3
Trial and improvement
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)