Simplifying Radicals

Perfect Squares

1

4

916

253649

64

81

100121

144169196

225

256

324

400

625

289

Look at these examples and try to find the pattern…

How do you simplify variables in the radical?

x7

1x x2x x3x x x4 2x x5 2x x x6 3x x

What is the answer to ? x7

7 3x x x

As a general rule, divide the exponent by two.

The remainder stays in the radical.

Simplifying variable radicands

• X²

• X

• X

• X

• X

4

16

25

100

144

=

=

=

=

=

8

20

32

75

40

=

= =

=

=

2*4

5*4

2*16

3*25

10*4

=

=

=

=

=

Perfect Square Factor * Other Factor

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AV

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ICA

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OR

M

48

80

50

125

450

=

= =

=

=

=

=

=

=

=

Perfect Square Factor * Other Factor

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ICA

L F

OR

M

87

249

5010

125

453

=

= =

=

=

=

=

=

=

=

Perfect Square Factor * Other Factor

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ICA

L F

OR

M

18

288

75

24

72

=

= =

=

=

=

=

=

=

=

Perfect Square Factor * Other Factor

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+To combine radicals: combine the coefficients of like radicals

Simplify each expression

737576

Simplify each expression

636556

547243

32782

Simplify each expression: Simplify each radical first and then combine.

323502

Simplify each expression: Simplify each radical first and then combine.

485273

229

22029

34*533*3

3*1653*93

18

288

75

24

72

=

= =

=

=

=

=

=

=

=

Perfect Square Factor * Other Factor

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Simplify each expression

636556

547243

32782

Simplify each expression

20556

32718

6367282

WORKSHEET

3) 6265)1 3334

5) 7) 105104 757

9) 2223243

11) 22618

13) 452535

15) 3327283

17) 33453122

4961284724

)19

8

462723723

)21

7

*To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals.

35*5

Multiply and then simplify

73*82

204*52

2

5

2

7

2

8

2

x

WORKSHEET(MULT)

5*5)1

WORKSHEET(MULT)

8*8)3

WORKSHEET(MULT)

3*3)5

WORKSHEET(MULT)

12*22)7

WORKSHEET(MULT)

62*6)9

WORKSHEET(MULT)

9*6)11

Using distributive Property

• a(b+c) = ab + ac

•

• a(b-c) = ab - ac

•

USING THE DISTRIBUTIVE PROPERTY

)23(15)13

USING THE DISTRIBUTIVE PROPERTY

)210(10)15

USING THE DISTRIBUTIVE PROPERTY

)55(5)17

USING THE DISTRIBUTIVE PROPERTY

)52(35)19

USING THE DISTRIBUTIVE PROPERTY

)52(35)19

USING THE DISTRIBUTIVE PROPERTY

24(10)21 )5

USING THE DISTRIBUTIVE PROPERTY

)1034(54)23

USING THE DISTRIBUTIVE PROPERTY

)52(35)19

Using the FOIL

)52)(35(

Using the FOIL

)38)(38(

Using the FOIL

)74)(75(

To divide radicals: divide the coefficients, divide the radicands if possible, and rationalize the denominator so that no radical remains in the denominator

7

56

7

6This cannot be

divided which leaves the radical in the

denominator. We do not leave radicals in the denominator. So

we need to rationalize by multiplying the

fraction by something so we can eliminate

the radical in the denominator.

42 cannot be simplified, so we are

finished.

This can be divided which leaves the

radical in the denominator. We do not leave radicals in the denominator. So

we need to rationalize by multiplying the

fraction by something so we can eliminate

the radical in the denominator.

10

5

This cannot be divided which leaves

the radical in the denominator. We do not leave radicals in the denominator. So

we need to rationalize by multiplying the

fraction by something so we can eliminate

the radical in the denominator.

12

3

This cannot be divided which leaves

the radical in the denominator. We do not leave radicals in the denominator. So

we need to rationalize by multiplying the

fraction by something so we can eliminate

the radical in the denominator.

20

5

Look at these examples and try to find the pattern…

How do you simplify variables in the radical?

x7

1x x2x x3x x x4 2x x5 2x x x6 3x x

What is the answer to ? x7

7 3x x x

As a general rule, divide the exponent by two.

The remainder stays in the radical.

Look at these examples and try to find the pattern…

How do you simplify variables in the radical?

2x x

4 2x x

6 3x x

As a general rule, divide the exponent by two.

2X

6Y

264 YXP

244 YX

10825 DC

=

=

=

=

=

125 x

26y

1087 nm

6102253 ba

2100c

=

=

=

=

=

3X =

=

XX *2

5Y

=

=

33YPX

2712 YX

9825 DC

=

=

= 5Y

PXYYX *22

5Y

PXYXY=