radical operations adding & subtracting radicals 1copyright (c) 2011 by lynda greene aguirre
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Radical Radical OperationsOperations
Adding & Subtracting Radicals
1Copyright (c) 2011 by Lynda Greene Aguirre
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Radical ExpressionsA RADICAL is the symbol best known as a
square root symbol.
Copyright © 2011 by Lynda Aguirre 2
A RADICAL EXPRESSION has radical terms and does not have an equal sign.
The object under the radical is called the RADICAND
Adding (& Subtracting)Terms with radicals can only be added
if their radicands are the same
These two terms have the same radicand: “3”
3Copyright (c) 2011 by Lynda Greene Aguirre
Addition: Same radicand
1. Add the coefficients
2. Bring down the
radical
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Subtraction: Same radicand
1. Subtract the
coefficients
2. Bring down the
radical
5Copyright (c) 2011 by Lynda Greene Aguirre
Addition and Subtraction: Same radicand
1. Add and Subtract
the coefficients
2. Bring down the
radical
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575152 Note: if there is no number in front of a radical, it is a “1”.
)712( 5
8 5
Different RadicandsSimplify 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)
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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
Rule: We can only add or subtract radicals with the same radicands, so try to simplify them first.
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9 and 4 are both perfect squares, so we can replace them with their square roots
39 24 1012)2(5)3(4
2
Different Radicands
Rule: We can only add or subtract radicals with the same radicand, so here, we can only combine the last 2 terms.
10Copyright (c) 2011 by Lynda Greene Aguirre
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7 and 3 are both prime numbers,
so we can’t simplify them any further.
3)45(72
3172
372
The “1” in front of the radical can be dropped
Different Radicands
These radicands cannot be reduced, so there is nothing that can be done to simplify this expression
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Solution
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Practice: Addition & Subtraction
See following slides for the step-by-step solutions
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Practice (key): Addition & Subtraction
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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 RadicalsRule:
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Example:
Then simplify if possible
Multiplying Radicals
Rule:
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Example:
Then simplify if possible
Distribute
Radicands are not the same, so this cannot be simplified further
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Multiplication of RadicalsMultiplication of Radicals
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Multiplication of RadicalsMultiplication of Radicals
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Another path to the same answer:
There are often several correct paths to the answer. Some are shorter than others.
Multiplication of RadicalsMultiplication of Radicals
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Expand the exponent to see the whole problem
Process: FOIL
Combine Like Terms and
Simplify Radicals
Multiplication of RadicalsMultiplication of Radicals
Multiplying Radicals
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73 xxMultiply using FOIL
2137 xxxx
Add the terms with the same
radicand 213)7( xxx
This is the solution
Practice Problems
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3763 263
x352 52152 x
273 xx 27763 xxx
231 x 2323 xx
Division of Radicals
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Dividing Radicals
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Rules: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
Short version of this:Multiply top and bottom
by the radicand.(This shortcut only works
for square roots)
Note: These are the same thing and can be changed as
needed
b
a
b
a
Dividing Radicals
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Reduce the outside numbers
Reduce the inside numbers
Rationalize the Denominator
4
18
9
3
3
12
9
23
9
Note: see “properties of radicals” notes for this “splitting the radical”
property
2
2
43
18
)2(3
29
6
23
Simplify the radical
Reduce the fraction
2
2
Dividing Radicals
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Rationalize the denominator
Simplify the top radical
34
182
3
3
34
182Terms with + or – signs
between them cannot be reduced separately
Distribute94
5432 Simplify the radicals
)3(4
6932
12
6332
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
Practice Problems
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34
92
2
3
2
532 2
1032
3
3 x3
33 x
Rationalize Denominator
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Rationalize the DenominatorRule: Radicals on the bottom of a fraction must be removed.
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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).
Root
Rationalize the DenominatorHigher 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.
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We only have 2 sevens
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
Power PowerPower
we need 1 more
to make 3.Use the rational
exponent property
Rationalize the DenominatorHigher 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.
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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.
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Rationalize the Denominator
For free math notes visit our website:
www.greenebox.com
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