Simplifying Radicals

Topic: Radical ExpressionsEssential Question: How are radical expressions represented and how can you solve them?

1 • 1 = 12 • 2 = 43 • 3 = 9

4 • 4 = 165 • 5 = 256 • 6 = 36

49, 64, 81, 100, 121, 144, ...

What numbers are perfect squares?

A List of Some Perfect Squares

1

4

916

253649

64

81

100121

144169196

225

256

324

400

625

289

4

16

25

100

144

= 2

= 4

= 5

= 10

= 12

That was easy!

8

20

32

75

40

=

= =

=

=

2*4

5*4

2*16

3*25

10*4

=

=

=

=

=

22

52

24

35

102

Perfect Square Factor * Other Factor

LE

AV

E I

N R

AD

ICA

L F

OR

M

48

80

50

125

450

=

= =

=

=

3*16

5*16

2*25

5*25

2*225

=

=

=

=

=

34

54

25

55

215

Perfect Square Factor * Other Factor

LE

AV

E I

N R

AD

ICA

L F

OR

M

1. Simplify 147

Simplify

1. .

2. .

3. .

4. .

2 18

72

3 8

6 236 2

+To combine radicals: combine the coefficients of like radicals

Hint: In order to combine radicals they must be like terms

Simplify each expression

737576 78

62747365 7763

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

323502

22

212210

24*325*2

2*1632*252

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

485273

329

32039

34*533*3

3*1653*93

Simplify each expression

636556

547243

32782

Simplify each expression

20556

32718

6367282

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

Hint: to multiply radicals they DO NOT need to be like terms

35*5 175 7*25 75

Multiply and then simplify

73*82 566 14*46

142*6 1412

204*52 1008 8010*8

2

5 5*5 25 5

2

7 7*7 49 7

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 8 2*4 22

Simplify

108

3

108

3

36

6

Uh oh…There is a

radical in the denominator!

Whew! It simplified!

Simplify

8 2

2 8

4 1

4

4

2

2

Uh oh…Another radical

in the denominator!

Whew! It simplified again! I hope they all are

like this!

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.

7

7*

7

6

49

42

7

42

42 cannot be simplified, so we are

finished.

Simplify

5

7

5

7

35

49 35

7

Since the fraction doesn’t reduce, split the radical up.

Uh oh…There is a

fraction in the radical!

How do I get rid of the radical in the

denominator?

Multiply by the “fancy one” to make the denominator a perfect

square!

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

2

2*

2

1

2

2

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

3

3*

12

3

36

33

6

33

2

3Reduce

the fraction.

2X

6Y

264 YXP

244 YX

10825 DC

= X

= Y3

= P2X3Y

= 2X2Y

= 5C4D5

3X

XX

=

=

XX *2

YY 45Y

=

= YY 2

Simplify

1. .

2. .

3. .

4. .

24 3x44 3x

2 48x448x

(3 6 2 3)(4 3)

3 6F

4 3 6O

3 2 3I

4 2 3L

3

12 6 3 32 2 8 3 2 3

12 6 9 2 8 3 6

Since there are no like terms, you can not combine.

Challenge:

How do you know when a radical problem is done?

1. No radicals can be simplified.Example:

2. There are no fractions in the radical.Example:

3. There are no radicals in the denominator.Example:

8

1

4

1

5