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
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