aim: simplifying radicals course: adv. alg. & trig. aim: how do i tame radicals? simply...

Aim: Simplifying Radicals Course: Adv. Alg. & Trig.

Aim: How do I tame radicals? Simply simplify!

Do Now:

Find the solution set and graph the inequalities

x2 – 7x + 10 < y and x2 – 7x + 10 < 0


Square Root

The square root of any real number is a number, rational or irrational, that when multiplied by itself will result in a product

that is the original number

525 The RadicalThe Radical

Square Root

Radicand

Radical sign

• Every positive radicand has a positive and negative sq. root.

• The principal Sq. Root of a number is the positive sq. root.

• A rational number can have a rational or irrational sq. rt.

• An irrational number can only have an irrational root.


Model Problems

Find to the nearest tenth:

62 130

53824 4153

= 7.9

= 232

Find the principal Square Root:

4

164

9225 529= +15 = +23

2

1

8

3

180c = 13.4

= 11.4

= 64.4

Simplify:

x2 = |x|

4x16 = 2x8

(x2 2x 1) = x + 1= x


Index of 2

anradical sign

radicand

index

of a number is one of the twoequal factors whose product is that number

Every positive real number has two square roots

The principal square root of a positive number k is itspositive square root, .

k

has an index of 2

k2 k

kIf k < 0, is an imaginarynumber

Square Root

Index of 2

81 9

81 9

2 ?


Cube Root

Index = 3

Index of 3

anradical sign

radicand

index

of a number is one of the threeequal factors whose product is that number

( k3 )( k3 )( k3 ) k

k3has an index of 3

273 3 273 3

principal cube roots


nth Root

The nth root of a number (where n is any counting number) is one of n equal factors

whose product is that number.

kn

k is the radicandn is the index

is the principal nth root of k

325 2 25 = 32

325 2 (-2)5 = -32

6254 5 54 = 625

6254 not real


Index of n

anradical sign

radicand

index

of a number is one of nequal factors whose product is that number

325 2 325 2

principal odd roots

6 664 2 64 not real principal even roots

has an index where n is any counting number

kn

nth Root

Index of n


Radical Rules!

T

22522550

True or False:

41664 6464

T

25100 1010

25

425425

88

T


simplified

Radical Rule #1

In general, for non-negative numbers a, b and n

Example:

494936 623

x2 x3

baba

x3 x2

x5 x3

x5 x3 = x4

x5

x8

Hint: will the index divide evenlyinto the exponent of radicand term?

n n na b a b

3 37 2x x = x33 9x3 7 2x x

4 2x x x x


Radical Rule #2True or False:

25

b4 5

b2

5

b2 25

b4T TIf and

25

b4 25

b4

Transitive Propertyof Equality

If a = b, and b = c, then a = c

a

b

a

b

In general, for non-negative numbers a, b, and nExample:

144

81

144

81

12

911

3

then

n

nn

a a

b b 4

625

256

4

4

625

256

5

4


Perfect Squares – Index 2

12

12 144

11

11 121

100

10

10

9

9 81

8

8 64

7

7 49

6

6 36

5

5 25

4

4 16

3

3 9

42

21

1 1


Perfect Square Factors

Find as many combinations of 2 factors whose product is 75

751

155

Find as many combinations of 2 factors whose product is 128

323 Factors that are Perfect Squares

642

168

253


Simplifying Radicals

baba

80 answer must be in radical form.Simplify:

80Find as many combinations of 2

factors whose product is 80

1 80

4 20

2 40

5 16

8 10

5 16

16 4

perfectsquare

4 5comes outfrom underthe radical

•To simplify a radical find, if possible, 2 factors of the radicand, one of which is the largest perfect square of the radicand.

•The square root of the perfect square becomes a factor of the coefficient of the radical.


Perfect Cubes

13 = 1

23 = 8

33 = 27

43 = 64

53 = 125

63 = 216

73 = 343

(x4)3 = x12

(-2y2)3 = -8y6


3 48 3 38 6

3 48 32 6


3 48 answer must be in radical form.Simplify:

3 48 3 8 6

2) Express the radical as the product of the roots of the factors

3) Simplify the radical containing the largest perfect power (cube)

1) Factor the radicand so that the perfect power (cube) is a factor



Simplify: 33

4

33

4

36

8

2) Express the radical as the quotient of two roots 3

3

6

8

3 6

2

1) Change the radicand to an equivalent fraction whose denominator is a perfect power.

33 2

4 2

3) Simplify the radical in the denominator

33

4


Model Problems

Simplify:20

KEY: Find 2 factors - one of which is the largest perfect square possible45

45 5225

124 324344344 38

8021 5162

151621

5421 52

96

12

16 6

4 3

96

12

4 6

2 3

2 6

32

6

3

2 2


Model Problems

Simplify:4 345a b

3 616x

2 2 29 5( )a b b

23 5a b b

2 33 8 2( )x

2 32 2x

2 2 29 5( )a b b

2 333 38 2 ( )x

aim: simplifying radicals course: adv. alg. & trig. aim: how do i tame radicals? simply...

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