Radical Operations
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4 7 8 3
Radical Operations
Adding & Subtracting Radicals
Copyright (c) 2011 by Lynda Greene Aguirre
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Radical Expressions
A RADICAL is the symbol best known as a
square root symbol.
A RADICAL EXPRESSION has radical
terms and does not have an equal sign.
The object under the radical is called
the RADICAND
Copyright © 2011 by Lynda Aguirre
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Adding (& Subtracting)
Terms with radicals can only be added
if their radicands are the same
These two terms have the same radicand: “3”
Copyright (c) 2011 by Lynda Greene Aguirre
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Addition: Same radicand
2. Bring
down the
radical
1. Add the
coefficients
Copyright (c) 2011 by Lynda Greene Aguirre
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Subtraction: Same radicand
2. Bring
down the
radical
1. Subtract
the
coefficients
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Addition and Subtraction: Same radicand
2 5 5 7 5
Note: if there is no number in front of a radical, it is a “1”.
2 5 1 5 7 5
1. Add and
Subtract
the
coefficients
( 2 1 7)
5
2. Bring
down the
radical
8 5
Copyright (c) 2011 by Lynda Greene Aguirre
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Different Radicands
Simplify terms with different radicands,
then add or subtract their coefficients.
Radicands are not the same so we
cannot add or subtract these
terms.
Try to simplify the terms
(see “simplify radicals” notes for more details)
Copyright (c) 2011 by Lynda Greene Aguirre
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Simplify the Radicals
NOW the radicands are the same so we can
add the coefficients
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Different Radicands
4 9 5 4
Rule: We can only add or subtract radicals with the
same radicands, so try to simplify them first.
9 and 4 are both perfect squares, so we can replace
them with their square roots
9 3
4 2
4(3) 5(2) 12 10
2
Copyright (c) 2011 by Lynda Greene Aguirre
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Different Radicands
2 7 5 3 4 3
Rule: We can only add or subtract radicals with the same
radicand, so here, we can only combine the last 2 terms.
2 7 (5 4) 3
7 and 3 are both
prime numbers,
so we can’t
simplify them
any further.
2 7 1 3
The “1” in
front of the
radical can
be dropped
2 7 3
Copyright (c) 2011 by Lynda Greene Aguirre
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Different Radicands
3 6 2
These radicands cannot be reduced, so there is nothing
that can be done to simplify this expression
3 6 2
Solution
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Practice: Addition & Subtraction
See following slides for the step-by-step solutions
Copyright (c) 2011 by Lynda Greene Aguirre
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Practice (key): Addition & Subtraction
Copyright (c) 2011 by Lynda Greene Aguirre
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Practice: Addition & Subtraction
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Practice: Addition & Subtraction
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Multiplication of
Radicals
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Multiplying Radicals
Rule:
Example:
Then simplify if possible
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Multiplying Radicals
Rule:
Example:
Distribute
Then simplify if possible
Radicands are not the same, so this cannot be simplified further
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Multiplication of Radicals
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Multiplication of Radicals
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Multiplication of Radicals
Another path to the same answer:
There are often several correct paths to the answer. Some are shorter than others.
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Multiplication of Radicals
Expand the exponent to
see the whole problem
Process: FOIL
Combine Like
Terms and
Simplify Radicals
Copyright (c) 2011 by Lynda Greene Aguirre
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Multiplying Radicals
Multiply using
FOIL
x 3 x 7
x x 7 x 3x 21
Add the terms
with the same
radicand
( x 7) x 3x 21
This is the solution
Copyright (c) 2011 by Lynda Greene Aguirre
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Practice Problems
3 6 7 3
2 5 3 x
3x 7 x 2
x 13 2
63 2
2 15 2 x 5
x 3 6x 7 x 7 2
3 x 2x 3 2
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Division of
Radicals
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Rules:
Dividing Radicals
Outside numbers on
the top can be divided
by Outside numbers
on the bottom.
Inside numbers on the top
can be divided by Inside
numbers on the bottom.
Reduce the
outside
numbers
Reduce the
inside
numbers
Radicals are not allowed
on the bottom
(denominator): see
“rationalizing the
denominator” notes for
more details on this
process
a
a
b
b
Note: These are
the same thing and
can be changed as
needed
Short version of this:
Multiply top and bottom
by the radicand.
(This shortcut only works
for square roots)
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Dividing Radicals
Reduce the outside
numbers
Rationalize the
Denominator
Simplify the
radical
3 18
9 4
1 9
3 2
9
3 2
2
2
Reduce the inside
numbers
Note: see “properties of
radicals” notes for this
“splitting the radical”
property
Reduce the
fraction
18
9 2 3 2
3(2)
6
3 4
Copyright (c) 2011 by Lynda Greene Aguirre
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2
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Dividing Radicals
2 18
4 3
Simplify the top radical
Terms with + or – signs
between them cannot
be reduced separately
Distribute
2
18 3
4 3
3
2 3 54
Rationalize the
denominator
Simplify the radicals
4 9
2 3 9 6
2
3
3
6
4(3)
12
This can only be reduced if the coefficients (outside numbers) could all be
divided by the same number. Since they can’t, this is the solution
Copyright (c) 2011 by Lynda Greene Aguirre
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Practice Problems
2 9
4 3
2 3 5
2
3x
3
3
2
2 3 10
2
3 x 3
3
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Rationalize
Denominator
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Rationalize the Denominator
Rule: Radicals on the bottom of a fraction must be removed.
Type 1: Single Term -Multiply the top and bottom by the same radical.
Type 2: Binomial (Two Terms)
-Multiply the top and bottom by the complex conjugate
(same thing, different signs).
Note: Don’t leave
a negative on the
bottom of a
fraction.
Move it in
front of the
fraction
and/or multiply
the top by it
(distribute).
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Rationalize the Denominator
Higher Order Radicals
To take out a radical, we must create a “perfect” number.
Recall that this means that the power must be divisible by the root.
Root
Root
Root
Power
Power
Power
If the power does not form a perfect number, multiply the top and bottom by
enough extra terms so that the powers will add up to a perfect number.
We only
have 2
sevens
we need
1 more
to make 3.
Use the rational
exponent
property
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Rationalize the Denominator
Higher Order Radicals
If the power does not form a perfect number, multiply the top and bottom by
enough extra terms so that the powers will add up to a perfect number.
We only
have 3 twos
we need 2
more
to make 5.
Use the
rational
exponent
property
Always check to see if you can reduce (cancel) or simplify
radicals when you reach the end of a problem.
Copyright (c) 2011 by Lynda Greene Aguirre
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Rationalize the Denominator
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For free math notes visit our website:
www.greenebox.com
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