My own exp nd radi
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Exponential &
Radicals KUBHEKA SN
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Exponential notation
represent as to the th power .
Exponent (integers)
Base (real
number)
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General case (n is any positive integers)
Special cases
Zero and negative exponent(where a c ≠ 0)
Example
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Law of Exponents
Law Example
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Theorem on negative Exponents
Prove:
Prove:
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Example :simplifying negative exponents
(1)
8
6
682
23242
234
9
3
)()()3
1(
)3
1(
x
y
yx
yx
yx
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Principal nth root Where n=positive integer greater than 1
= real number
Value for Value for
= positive real number b
Such that
=negative real number b
Such that
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Properties of:
RADICALradican
d
index
Radical sign
PROPERTY EXAMPLE
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Example:combining radicals
Question:
12 5
125
125
32
41
41
3 2
4
1
1
32
α
α
αα
α
α
α
α
)(
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Law of Radicals
law example
WARNING!
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Simplifying RadicalsOperations with
Radicals
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Review - Perfect Squares
2
2
2
2
2
2
1 1
2 4
3 9
4 16
5 25
6 36
1 1
4 2
9 3
16 4
25 5
36 6
2
2
2
2
2
2
7 49
8 64
9 81
10 100
11 121
12 144
49 7
64 8
81 9
100 10
121 11
144 12
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Rules for Radicals
21) a a
b) a2 ba
3) a
b b
a
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Simplifying Square Roots
Simplify:
Step 1Look for Perfect Squares (Try to use the largest perfect square possible.)
Step 2Simplify Perfect Squares
Step 3Multiply the numbers inside and outside the radical separately.
48
3 16
43
4 3
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If you miss the largest
perfect square, it will
just take more steps.
Simplify: 48
4 12
34
2 2 3
4 3
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Variables2a a
2x xAny even power is a perfect square.
4 2
10 5
90 45
x x
x x
x x
The square root exponent is half
of the original exponent.
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Odd powers
When you take the square root of an odd power, the result is always an even power and one variable left inside the radical.
5 2
11 5
91 45
x x x
x x x
x x x
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Simplifying using variables
When you simplify an even power of a variable and the result is an odd power, use absolute value bars to make sure your answer is positive.
14 7
14 12 7 6
x x
x y x yEven
powers do not need absolute value.
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Simplifying Numbers & VariablesSimplify: 316x
Step 1Pull out perfect squares
Step 2Simplify
16 2x x
x4 x
4x x
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Radical Multiplication
a ab b You can only multiply radicals by other radicals
8 3Both under the radical
CAN multiply
8 3Not under the radical
CANNOT multiply
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What is an “nth Root?”
Extends the concept of square roots.
For example:
A cube root of 8 is 2, since 23 = 8
A fourth root of 81 is 3, since 34 = 81
For integers n greater than 1, if bn = a then b is an nth root of a.
Written where n is the index of the radical.
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Rational Exponents
nth roots can be written using rational exponents.
For example:
In general, for any integer n greater than 1.
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Real nth Roots If n is odd:
a has one real nth root
If n is even:
And a > 0, a has two real nth roots
And a = 0, a has one nth root, 0
And a < 0, a has no real nth roots
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Finding nth Roots
Find the indicated real nth root(s) of a.
Example: n = 3, a = -125
n is odd, so there is one real cube root: (-5)3 = -125
We can write
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Your Turn!
Solve each equation.
5x4 = 80
(x – 1)3 = 32
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http://www.slideshare.net/nurulatiyah/radical-and-exponents-2?qid=b15cb847-ee58-4b34-aaba-ce8e8ab498a5&v=default&b=&from_search=10
http://www.slideshare.net/holmsted/roots-and-radical-expressions?qid=b15cb847-ee58-4b34-aaba-ce8e8ab498a5&v=default&b=&from_search=12
http://www.slideshare.net/hisema01/71-nth-roots-and-rational-exponents?qid=b15cb847-ee58-4b34-aaba-ce8e8ab498a5&v=default&b=&from_search=15