Operations on

RadicalsModule 14 Topic 2

Table of Contents

Slides 3-5: Adding and Subtracting Slide 5: Simplifying Slides 7-10: Examples Slides 11-14: Multiplying Slides 15-16: Conjugates Slides 17-19: Dividing Slides 20-22: Rationalizing the Denominator Slides 23-28: Practice Problems

Audio/Video and Interactive Sites

Slide 7: Gizmo Slide 25: Interactive

Addition and Subtraction

Adding and Subtracting radicals is similar to adding and subtracting polynomials.

Just as you cannot combine 3x and 6y, since they are not like terms, you cannot combine radicals unless they are like radicals.

• If asked to simplify the expression 2x + 3x, we recognize that they each share a common variable part x that makes them like terms, hence we add the coefficients and keep the variable part the same. xxx 523

• Likewise, if asked to simplify radicals. If they have “like” radicals then we add or subtract the coefficients and keep the like radical. Two radical expressions are said to be like radicals if they have the same index and the same radicand.

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• If the radicals in your problem are different, be sure to check to see if the radicals can be simplified. Often times, when the radicals are simplified, they become the same radical and can then be added or subtracted. Always simplify first, if possible.

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Recall: Simplifying Radicals235998 zyx

zzyyyxxxxx 1198Divide the number under the radical.If all numbers are not prime, continue dividing.

zzyyyxxxxx 11338

Find pairs, for a square root, under the radical and pull them out.

zyxx

zzyyyxxxxx

3

11338

Multiply the items you pulled out by anything in front of the radical sign.

Multiply anything left under the radical . xyzyxx 1138

xyyzx 1124 2It is done!

Simplify: 123

323432212

33

323

123

Now you can simplify by using like terms.

*These are not like terms, however the 12 can be simplified.

Gizmo: Operations

with Radicals

Example:

Example:

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DO NOT ADD THE NUMBERS UNDER THE RADICAL!

You may not always be able to simplify radicals.

Since the radicals are not the same, and both are in their simplest form, there is no way to combine them. The answer is the same as the problem:

3274 Simplify

3274 Answer:

Example:

Example:

Example:

*(Hint: P = 2L + 2W)

44 2 62 2 xx 4 28 x

33 10 53 4 xx 33 10 53 4 xx

form?

radicalsimplest in garden theofperimeter theisWhat

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*Neither radical can be simplified. The expression is already in simplest form.

Multiplying Radicals

• To multiply radicals, consider the following:

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and

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33

33

and

Multiplying Radical Expressionnnn abba

Multiplying RadicalsYou can simplify the product of two radicals by writingthem as one radical then simplifying. Here is an example of how you can use this Multiplication property to simplify a radical

expression.

Example:3 83 7 67 xx 3 1542x

3 555327 xxx

35 42 x

3 8767 xx

Example: xyyx 1515 3 24225 yx

2221515 yxx

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Example: Multiply )77(7 xx

*Recall that to multiply polynomials you need to use the Distributive Property.

(a + b)(c – d)=a(c – d) + b(c – d)=ac – ad + bc - bd)77(7 xx

xx 4977 2

xx 777

xx 497

7

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You should recognize these from Topic 1 Notes

You should also remember the rules of exponents, when you multiply terms with the same base, you add exponents. When you divide, you subtract exponents.

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Putting the above information together, you should see that these rules can be combined to solve problems such as those below.

14

97

1

2

1

7

1

2

17 )1 xxxxxx

20

114

1

5

4

4

1

5

4

4

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x

x

x 2) xx

x

Example:

)26)(26(

)26(2)26(6

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436

4

26

Notice that just like a difference of two squares, the middle terms cancel out.

We call these conjugate pairs.

The conjugate of .

They are a conjugate pair. When we multiply a conjugate pair, the radical cancels out and we obtain a rational number.

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Distribute (2 ways to do so)

Conjugates

)2(2)6(2)2(6)6(6

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436

4

26

Simplified Radical Expressions are recognized by…

No radicands have perfect square factors other than 1.

No radicands contain fractions.

No radicals appear in the denominator of a fraction.

Dividing Radicals• To divide radicals, consider the following:

244

16

4

16

22

4

4

16

and

288

64

8

64

22

4

8

64

333

3

3

3

and

Dividing Radical Expressions

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n

b

a

b

a

Dividing RadicalsYou can simplify the quotient of two radicals bywriting them as one radical then simplifying.

Example:x

x

2

90 181745x

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5x 3 8x

x

x

2

90 18

Dividing Radicals

Example:42

98

5

6

yx

yx

5

6

5

6 2356 yyxyx

Notice that after simplifying the radical, we still have a square root in the denominator.

We have to find a way to get rid of the radical.

This is called rationalizing the denominator.

Case I: There is ONE TERM in the denominator and it is a SQUARE ROOT.

When the denominator is a monomial (one term), multiply both the numerator and the denominator by whatever makes the denominator an expression that can be simplified so that it no longer contains a radical. In this case it happens to be exactly the same as the denominator.

2

7

2

14

4

14

2

2

2

7

Rationalizing the Denominator.

Case II: There is ONE TERM in the denominator, however, THE INDEX IS GREATER THAN TWO.

Sometimes you need to multiply by whatever makes the denominator a perfect cube or any other power greater than 2 that can be simplified.

3

3

11

9

11

1089

1111

1111

11

9 3

3

3

3

3

Case III: There are TWO TERMS in the denominator. We also use conjugate pairs to rationalize denominators.

63

63

63

63

63

63

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63

)33(23

321 3

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Be sure to enclose expressions with multiple terms in ( ). This will help you to remember to FOIL these expressions. Always reduce the root index (numbers outside radical) to the simplest form (lowest) for the final answer.

Application/Critical ThinkingA. Find a radical expression for the perimeter

and area of a right triangle with side lengths .38 12, ,34

Perimeter … (P = a + b + c)

3 12123 8123 4

bh 2

1 Area Hint: In a

right triangle, the longest side is the hypotenuse of the triangle.

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)12)(34(

A

A

86.1338

This is longer than 12, so 12 is a leg.

Application/Critical ThinkingB. The areas of two circles are 15 square cm and

20 square cm. Find the exact ratio of the radius of the smaller circle to the radius of the larger circle.

Find the radius of each first . . . A = π r2

Smaller Circle

11

15

15 2

r

rLarger Circle

11

20

20 2

r

r2

3

4

3

20

15

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Practice Problems

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Practice Problems and Answers

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