6-1 nth roots reg
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6-1 Nth Roots Objective: To simplify radicals having various indices, and to use a calculator to estimate the roots of numbers.
Transcript of 6-1 nth roots reg
6-1 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
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
Roots of negative numbers*
Even roots: Negative numbers have no even roots. (undefined)
Odd Roots: Negative numbers have negative roots.
3 27
4
undefined
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
if n (index) is an even integer
a<0 has no real nth roots
a=0 has one real nth root
a>0 has two possible real nth roots
if n is an odd integer
a<0 has one real nth root
a=0 has one real nth root
a>0 has one real nth roots
solution) real a(not 42 16 i 23 8
04 0 03 0
4 224 32 324
x 33 27
MORE EXAMPLES
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
•
