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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

8

20

32

75

40

=

= =

=

=

2*4

5*4

2*16

3*25

10*4

=

=

=

=

=

22

52

24

35

102

Perfect Square Factor * Other FactorL

EA

VE

IN

RA

DIC

AL

FO

RM

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

ICA

L F

OR

M

Simplify32

216

4

24

Simplify 45

59

3

53

Simplify 96

16 6

64

4

2 3 6

Simplify

216

4 54

66

69OR

6

216

36 6

66

Simplify each expression

737576 78

62747365 7763

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

323502 2*1632*252

22

212210

24*325*2

Now you have like terms to combine

Not like terms, they can’t be combined

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

485273 3*1653*93

329

32039

34*533*3

Now you have like terms to combine

Not like terms, they can’t be combined

1. multiply the coefficients 2. multiply the radicands 3. simplify the remaining

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

2

8 8*8 64 8

2

x xx * 2x x

Short cut

Squaring a Square Root

Short cut

2

6222 62 2464

2

53 22 53 4559

2

5

3

2

2

5

3

25

3

Squaring a Square Root

-divide the coefficients

-rationalize the denominator so that no radical remains in the

denominator

63

26

There is an agreement

31

in mathematics that we don’t leave a radical

in the denominator of a fraction.

So how do we change the radical denominator of a fraction?

31

(Without changing the value of the fraction) The same way we change the denominator of any fraction…

41

12

3

3

3

4

1

For Example:

Multiply by a form of 1.

By what number can we multiply

to change to a rational number? 3

13

The answer is . . . . . . by itself!

3

1

3 3 23 3

3

3

33

31

3

3

Squaring a Square Root gives the Root!

3

133

Because we are changing the denominator

we call this process rationalizing.

to a rational number,

2

4

2224

Rationalize the denominator:

2

24

(Don’t forget to sim

plify)22

2

2

2

2

4

12

64

128

1212128

Rationalize the denominator:

36

3

1296

(Don’t forget to sim

plify)

(Don’t forget to sim

plify)

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

7

568

2*4 22

Simplify.7

56

Simplify.

Simplify.

108

3

108

3

366

Uh oh…There is a

Whew! It simplified!

Simplify

8 2

2 8

4 1

4

4

2

2

Uh oh…Another

Whew! It simplified again! I hope they all are like this!

Simplify

5

7

5

7

75

7 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 1” to make the denominator a

perfect square!

*

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

7

7*

7

6

49

42

7

42

42 cannot be simplified, so we are

finished.

Fractional form of “1”

10

5

2

2*

2

1

2

2

Simplify fraction

Rationalize Denominator

2

1

12

3

3

3*

12

3

36

33

6

33

2

3Reduce the

fraction.

Use any fractional form of “1” that will result in a perfect

square

Finding square roots of decimals

If a number can be made be dividing two square numbers then we can find its square root.

For example,

= 3 ÷ 10

= 0.3

0.09 = 9 ÷ 100

Find 0.09

= 12 ÷ 10

= 1.2

1.44 = 144 ÷ 100

Find 1.44

If a number cannot be written as a product or quotient of two square numbers then its square root cannot be found exactly.

Use the key on your calculator to find out 2.

The calculator shows this as 1.414213562

This is an approximation to 9 decimal places.

The number of digits after the decimal point is infinite.

Approximate square roots

Estimating square roots

What is 10?

10 lies between 9 and 16.

Therefore,

9 < 10 < 16

So,

3 < 10 < 4

Use the key on you calculator to work out the answer.

10 = 3.16 (to 2 decimal places.)

10 is closer to 9 than to 16, so 10 will be

Suppose our calculator does not have a key.

36 < 40 < 49

So,

6 < 40 < 7

6.32 = 39.69 too small!

6.42 = 40.96 too big!

Trial and improvement

Find 40 40 is closer to 36 than to 49, so 40 will be about 6.3

6.332 = 40.0689 too big!

6.322 = 39.9424 too small!

Suppose we want the answer to 2 decimal places.

6.3252 = 40.005625 too big!

Therefore,

6.32 < 40 < 6.325

40 = 6.32 (to 2 decimal places)

Trial and improvement