Exponential &
Radicals KUBHEKA SN
Exponential notation
represent as to the th power .
Exponent (integers)
Base (real
number)
General case (n is any positive integers)
Special cases
Zero and negative exponent(where a c ≠ 0)
Example
Law of Exponents
Law Example
Theorem on negative Exponents
Prove:
Prove:
Example :simplifying negative exponents
(1)
8
6
682
23242
234
9
3
)()()3
1(
)3
1(
x
y
yx
yx
yx
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
Properties of:
RADICALradican
d
index
Radical sign
PROPERTY EXAMPLE
Example:combining radicals
Question:
12 5
125
125
32
41
41
3 2
4
1
1
32
α
α
αα
α
α
α
α
)(
Law of Radicals
law example
WARNING!
Simplifying RadicalsOperations with
Radicals
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
Rules for Radicals
21) a a
b) a2 ba
3) a
b b
a
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
If you miss the largest
perfect square, it will
just take more steps.
Simplify: 48
4 12
34
2 2 3
4 3
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.
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
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.
Simplifying Numbers & VariablesSimplify: 316x
Step 1Pull out perfect squares
Step 2Simplify
16 2x x
x4 x
4x x
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
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.
Rational Exponents
nth roots can be written using rational exponents.
For example:
In general, for any integer n greater than 1.
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
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
Your Turn!
Solve each equation.
5x4 = 80
(x – 1)3 = 32
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
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