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Transcript of Bellwork. Survey results: Students who voted for online homework: 84% Students who voted for paper...
Bellwork
1. Solve
2. Solve
Survey results:
Students who voted for online homework: 84% Students who voted for paper homework: 16%
Students who wants to keep group testing: 52% Students who prefer individual testing: 48%
Properties of Exponents
Section 6.1 and 6.2
What will we learn?
You Will Learn Use zero and negative exponents.
Use the properties of exponents. Solve real-life problems involving exponents.
Exponential Notation
an = a * a * a * a…* a (where there are n factors)
The number a is the base and n is the exponent.
Zero and Negative Exponents
If a ≠ 0 is any real number and n is a positive integer, thena0 = 1a-n = 1/an
You try
BellworkSimplify the expressions;
Laws of Exponents Product of power property When multiplying two powers of the same base, add
the exponents.aman = am+n
Quotient of power property When dividing two powers of the same base, subtract
the exponents. am/ an = am – n
Power of a power properties When raising a power to a power, multiply the
exponents. (am)n = amn
Example – Product Property
(-5)4 * (-5)5 = (-5)4+5 = (-5)9 = -1953125
Example
x5 * x2 = x5+2 = x7
Example – Neg. Exponent
(-5)-6(-5)4 = (-5)-6+4 = (-5)-2 =
251
251
Example – Quotient of Powers
3
5
xx 35x 2x
Example – Quotient of Powers
10
5
xx 105x 5x 5
1x
Example – Power of a Power
(23)4 = 23*4 = 212 = 4096
Example
(34)2 = 34*2 =38 = 6561
Bellwork
Laws of Exponents (ab)n = anbn
When raising a product to a power, raise each factor to the power.
(a/b)n = an / bn
When raising a quotient to a power, raise both the numerator and denominator to the power.
(a/b)-n = (b/a)n
When raising a quotient to a negative power, take the reciprocal and change the power to a positive.
a-m / b-n = bn / am
To simplify a negative exponent, move it to the opposite position in the fraction. The exponent then becomes positive.
Example – Zero Exponent
(7b-3)2 b5 b = 72 b-3*2 b5 b = 49 b-6+5+1 = 49b0 =49
Example – Power of Quotient
2
5sr
25
2
sr 10
2
sr 102sr
3
yx
3
3
yx
4
7
xx
1
47x 3x
7
5
xx
57
1x 2
1x
Basic Examples
Scientific Notation
Scientific Notation—shorthand way of writing very large or very small numbers.4 x 1013
4 and 13 zero’s1.66 x 10-12
0.00000000000166
Scientific Notation
131,400,000,000= 1.314 x 1011
Move the decimal behind the 1st number
How many places did you have to move the decimal?
Put that number here!
Example – Scientific Notation
131,400,000,000 = 5,284,0001.314 x 1011 =5.284 x 106
61110*284.5314.1 900,2410*249. 5
Bellwork
1. Simplify the following:
2. Simplify the following:
3. Simplify the following:
223 73 xyzzyx
32 238 xyxyxy
3
2
3
35abba
Finding nth RootsYou can extend the concept of a square root to other types of roots.
For example, 2 is a cube root of 8 because = 8, and 3 is a fourth root of 81 because = 81.
In general, for an integer n greater than 1, if = a, then b is an nth root of a. An nth root of a is written as , where the expression n √ a is called a radical and n is the index of the radical. You can also write an nth root of a as a power of a.
nth root
If n is any positive integer, then the principal nth root of a is defined as:
If n is even, a and b must be positive.
means nn a b b a
If you assume the Power of a Power Property applies to rational exponents, then the following is true.
Examples:
36 1. 3612 6
643 2. 6413 4
149
12
3. 4912
149
83
4. 1
8
13 2
Examples:
17
8 13
Rational Exponents
For any rational exponent m/n in lowest terms, where m and n are integers and n>0, we define:
If n is even, then we require that a ≥ 0.
/ nm n ma a
Properties of nth roots
if n is odd
| | if n is even
n n n
nn
n
m n mn
n n
n n
ab a b
a ab b
a a
a a
a a
Rationalizing the Denominator
We don’t like to have radicals in the denominator, so we must rationalize to get rid of it.
Rationalizing the denominator is multiplying the top and bottom of the expression by the radical you are trying to eliminate and then simplifying.
More Examples
43 72 aa 4372 a 714a 232 285 rrr 232285 r 780r
33xy 3333 yx 3327 yx
2
32ba
22
22
32ba
2
2
94ba
3522 nm 3532312 nm 15632 nm 1568 nm
xx
28 4
128 14x 34x
5
3
39zz
35
139z
2
13x 2
3x
More Examples
223 73 xyzzyx 21121373 zyx 33421 zyx
32 238 xyxyxy 312111238 yx 6348 yx
22232 23 xyyx 222121232221 23 yxyx
4264 49 yxyx 462449 yx 10636 yx
3
2
3
35abba
323131
313331
35
baba
633
393
35
baba
63
39
27125
baba
36
39
27125ba
3
6
27125ba