Drill #37Factor the following polynomials

Simplify.

Find the value of r:

rr

r

zyxzyx

xx

n

2

5

35432332

2

.3

)3()2(.2

126.1

Drill #38Factor the following polynomials

Simplify.

3 63

42

8.3

25.2

1000.14

yx

yx

xx

Drill #60Factor the following polynomial

Evaluate the following roots:

443.1 2 xx

100.4

27.3

25.2

3 6

2

y

x

7-4 Nth RootsObjective: To simplify radicals

having various indices, and to use a calculator to estimate the roots of

numbers.

Square Roots

What power is a square root?

A square is the inverse of a square root…

?33

Square Root*

Definition: For any real numbers a and b, if

then a is a square root of b or

We can also write square roots using the ½ power.

ba 2

ab

bb 2

1

Cube Root*

Definition: For any real numbers a and b, if

then a is a cube root of b or

We can also write cube roots using the 1/3 power.

ba 3

ab 3

33

1

bb

nth Root*

Definition: For any real numbers a and b, if

then a is a nth root of b or

We can also write nth roots using the power.

ban abn

nn bb 1

n

1

How many ways can you …

Multiply two numbers to get a positive #?

Multiply two numbers to get a negative #?

Examples: Roots (of powers of 2)Even Roots: Odd Roots:

2256

264

216

24

8

6

4

2512

2128

232

28

9

7

5

3

Principal Root*Definition: When there is more than one real

root of a number (even numbered roots), the non-negative root is the principal root

Principal Root

Negative Root

Both864

864

864

Examples: Roots (of powers of 2)

Both PrincipalNegative

Roots: Roots: Roots:

2256

264

216

24

8

6

4

2256

264

216

24

8

6

4

2256

264

216

24

8

6

4

Roots of negative numbers*

Even roots: Negative numbers have no even roots. (undefined)

Odd Roots: Negative numbers have negative roots.

327

43

undefined

Examples: Roots (of powers of 2)Even Roots: Odd Roots:

.256

.64

.16

.4

8

6

4

undef

undef

undef

undef

2512

2128

232

28

9

7

5

3

Roots: Number and Types

Even Roots Odd Roots

Positive 2 (one positive, one negative)

1 (positive)

Negative 0 (undefined) 1 (negative)

464.64

464864

3

3

undef

Even Roots (of variable expressions)*

When evaluating even roots (n is even) use absolute values (if resulting power is odd).

.

.25

3)3(

2)2(

4 824 8

424

4 434 12

22

undefzzz

undefyyy

xx

aa

Odd Roots (of variable expressions)*

When evaluating odd roots (n is odd) do not use absolute values.

232

7243

535 15

33 3

aa

aa

Evaluating Roots of Monomials

To evaluate nth roots of monomials:

(where c is the coefficient, and x, y and z are variable expressions)

or

• Simplify coefficients (if possible)• For variables, evaluate each variable separately

nnnn

nnnnn

zyxc

zyxccxyz1111

)()()()(

Evaluating Roots of Monomials*To find a root of a monomial

• Split the monomial into a product of the factors, and evaluate the root of each factor.

• Variables: divide the power by the root Coefficients: re-write the number as a product of prime numbers with powers, then divide the powers by the root.

325 535 525 55 1510

424288

2)()(232

7)(74949

yxyxyx

xxxx

Use a calculator to find the roots*

Find the following roots using a calculator (round to the 3 decimal places):

5

4

674.3

275.2

150.1

ex

ex

ex

Examples: Simplify

16

10

84

5 2520

82

4

)7(:2

36:1

16.

243.

)6(.

16.

ycw

xcw

yxd

bac

xb

ya

Examples: Simplify

4 12

6

6 182

4 4

)12(16.2

36:1

)3(64.

.

xcw

ycw

xb

ya