Chapter 9 Continued – Radical Expressions and EquationsMultiplication & Division of Radical Expressions

When multiplying radical expressions, you can just put everything that’s under a radical sign together under one big radical sign.

Distributive Property:

BE CAREFUL!

xyyx

yyxx

yx

xyyxx

xyyxx

Multiply

3

26

37

224

224

6

36

36

623

623

:

xxx

xxxx

xxx

22

)2)(2(2

22

confusion. avoid to

2 as written be should 2

2 equalnot does 2

xx

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2not is

of outside on the 2 thesince

2 equalnot does 2

,

x

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Also

Example 3 Multiply

yxyx

yxyx

yxyxyx

yyxyyxxx

yxyx

253Multiply

:one isYou try th

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2 5 4 10

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FOIL Use

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Example Multiply

Notice that when ra adical expression has two terms, all radicals disappear when you multiply the expression by its conjugate.Try this one:

other.each of conjugates called are and )(

:squares twoof difference theof case factoring special the was that thisNotice

494

)7)(7(272722

7272

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(a-b)ba

-bab)(a-b)(a

x

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xx

11 xx

Radical Expressions in Simplest Form

A radical expression is in simplest form if:

1.The radicand contains no factor greater than 1 that is a perfect square.2.There is no fraction under the radical sign.3.There is no radical in the denominator of a fraction.

is not in simplest form because there is a fraction under the radical sign. This can be simplified by taking the square root of the numerator and the denominator.

36

2

6

2 244

z

x

z

x

z

x

b

a

b

a Roots Square ofProperty Quotient

xy

yx

xy

yx 22 44Simplify

x4

x2

3

2 Is not in simplest form because there is a radical expression in the denominator;The way to simplify is to multiply both numerator and denominator by 3

3

32

3

3

3

2

This doesn’t always work when there is a two-term expression with at least one radical term added to another term.

96

32

3

3

3

22

yy

yy

y

y

y

y

3

3

3

2

y

y

y

y

The trick for these types is to multiply the numerator and denominator by the conjugate.

22

2

3

232

y

yy

9

232

y

yy

33

32

yy

yy

SIMPLIFIED!

UGH!

Solving Equations Containing Radical Expressions

Property of Squaring Both Sides of an Equation

If a and b are real numbers and a=b, then a2=b2

TRUE 523

529

523(3)

:Check

3

93

33

sign. radical under the fromout get x tosidesboth Square

33

523x :Solve

22

x

x

x

x

It’s very important to check your solution because some “solutions” actually make the original equation untrue.

Example:

Notice that when you get the constants on one side, your equation says that the radical expression must equal a negative number.This is impossible! Therefore there is NO SOLUTION to an equation like this.

9

312

x

x

55

525

5)5(30

5Check

6)6(30

6Check

5or 6

)5)(6(0

300

30

30

:

2

2

x

x

xx

xx

x-x

x-x

xx

Example

square both sides

This is now a degree 2 equation so put it in standard form, factor it, then use zero-product rule.

Impossible because the principal square root of a number can never be negative. Therefore -6 is not a possible solution.

OK

Therefore, only solution is {5}

You try!

Solve:

a =

Solve equation and exclude any extraneous solutions:

m =

Solve:

123

143

1599

:

9

436

20416

)5(416

524

5240

5521

5521

again. process start thethen

termlike Combine right. on the radical a have still

but weleft on the radical theof ridgot This

5521

51

51

sides.both squaring before

other each of oppositeson sexpression radical put the case In this

15

22

22

CHECK

x

x

x

x

x

x

xxxxx

xxx

xxx

xx

xx

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