radical operations adding & subtracting radicals 1copyright (c) 2011 by lynda greene aguirre

Radical Radical OperationsOperations

Adding & Subtracting Radicals

1Copyright (c) 2011 by Lynda Greene Aguirre

36 3874

Radical ExpressionsA RADICAL is the symbol best known as a

square root symbol.

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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”

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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

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Addition and Subtraction: Same radicand

1. Add and Subtract

the coefficients

2. Bring down the

radical

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57552

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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4594

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.

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343572

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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263

Solution

263

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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