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An exponent tells how many times a number is multiplied by itself.
83Base
Exponent
#1
Factored Form
3 • 3 • a • a • a
Exponential Form
32a
3
x • y • x • 2 2x
2y
1
8•8•8 = 512
22 3
2 23 3 4
9
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# 2When simplifying exponents you must watch the sign and the parenthesis!
52
= 5•5 = 25
–52
= –5•5 = –25
(-5)2
=(-5)(-5) = 25 –(5)2
= - (5)(5) = –251
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copyright©amberpasillas2010
Evaluate The Power#3
1)
2)
3)
4)
25
42
310
212
5 5 25
2 2 2 2 16
10 10 10 1000
12 12 144
To find 5 on my calculator I type in
4
5 4^ = 625
5 yx 4 = 625
Try to find 9 = 7293
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Powers of Ten
210
10
10
10
10
3
4
5
100
1,00010,000
100,000
10110
110
-1 = 0.1
# 4
10-2
1
110 2
1100
= 0.01
10-3 1
10 3 11000
= 0.001
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Negative Exponents
# 5
EXAMPLES:
232
1
3
4( 5) 4
1
( 5)
For any integer n, a-n is the reciprocal of an
1nn
aa
A negative exponent is an inverse!
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Any number to the zero power is ALWAYS ONE.
x0 = 1
Ex:
# 6
04 12 25 5 2 25 05 1
03 4 1 1
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Exponents and Parenthesis
#7Factored Form
8 • x • x • xExponential Form
8x3
4(xy)(xy) 4(xy)2
(8x)3 (8x)(8x)(8x) = 8
3x
3
= 4 x2y
2
(5x3)
2 (5 x x x)(5 x x x) = 25x6
(2y2z)
2 (2 y y z) (2 y y z) = 22y
4z
2
= 512x3
= 52(x
3)2
= 4y4z
2
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Fractions With Exponents
311) 2
222) 3
214) 3
225) 5
26) 9
12 2 2
2 23 3
231
9
252
254
219
181
18
49
# 8
213) 5
1 15 5 1
25
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Negative Exponent Examples51) n
3 42) a b
44) 3a
45) 3a
26) 5x 3 03) m n
51n
3 41=
a b
413a
43a
41
3a 4
181a
215x
25
x
31 1
m
# 9
31
m
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21
8
Just flip the fraction over to make the exponent positive!
28
1
2
2
8
1
#10
64
24
7
27
4
2
2
7
4 49
16
31
4
34
1
3
3
4
( 1)
64
1
64
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When multiplying powers with the
same base, just ADD the exponents.
For all positive integers m and n:am • an = am + n
Ex :
#11
(32)(3
3) = (3 • 3) • (3 • 3 •3)
= 32+3 = 3
5
(x5)(x
4) = x
5+4 = x9
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To find the power of a power, you MULTIPLY the
exponents . This is used when an exponentis on the outside of parenthesis.
# 12
= 53a
2•3b
3 (51a
2b)
3
(21x
3)5
= 25x
3•5
8(31y
8z)
2 = 8 (3
2y
8•2z
2)
= 125a6b
3
= 32x15
= 72y16
z2
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#13
= x8+5
x8• x
5
(4a7b)
3 = 43a
7•3b
3
= x13
= 64a21
b3
Product of a Power Property
Power of a Power Property
= (-12)2(-3 • 4)
2
=144
Power of a Product Property
= (-12)(-12)
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Prime Factorization is when you write a
number as the product of prime numbers. Factor Tree 36
#14
36 2 2 3 3 2 236 2 3
2 18
2 9
3 3
Circle the
prime numbers
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Factoring#15
1)410m
15m=
2 5 m m m m 3 5 m =
32m
3
2)3
2
27a b
36ab=
3•3•3• • • •a a a b2• 2 •3•3• • •a b b =
23
4
a
b
12
2 6
2 3
2 2 3
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# 16When Dividing Powers with the same base,
just SUBTRACT THE EXPONENTS. This is called the
Quotient of Powers Property.
x 5
x 2 =x x x x x
x x = x x x 3x
x5
x 2 =5 2x 3x
or
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#17
2 315x y12xy
1512
2 1x 3 1y
54x 2y
2 315x y
12xy
3
3
5
4
21
25xy
4
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COEFFICIENT: The number in front of the variable is the coefficient. Multiply coefficients. Add exponents if the bases are the same
3 52x 4x 8 3 5 x 8 8x
# 18
2(7ab)(2a )
14
1) 2) 2 3 4(2x y )( 3x y)
63a b 6x 4y
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# 19 Dividing Powers With Negatives
Quotient of Powers Propertyx a
x b =a bx
3
4
6x
8x =
=3
4
6x
8x
3
4
3x
4x =3 43x x
4
=
73x
4
3
4
3x
4x =3 43
x4
=
73x
4
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3 42x 5x
2 5 3 4x 710x
#20
Simplify.5 71) (8a ) (3a )
72) (-3a) (4a )
2 3 53) (9x y )(-2xy )
2 3 54) (6a bc )(5ab )
24 12a
12 8a
18 3x 8y
30 3a 6b 3c
1
1
11
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# 21 Quotient of Powers Property
Quotient of Powers Propertyx a
x b =a bx
2
377
2 37 57 5
17
2 37 7
32
1 77
2 31 17 7
5
17
Same
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Extras
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Exponents & PowersAn exponent or power tells how manytimes a number is multiplied by itself.
34
BaseExponent
#1
25 “Five to the 2nd power” 5 5“Five squared”
37 “Seven to the 3rd power” 7 7 7 “Seven cubed”
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Multiplying Powers: If bases are the same add exponents.
7 4 x x 7 4x 11xPower of a Power: Used when exponents are on the outside of parenthesis, just multiply exponents.
2 4 3(2a b )43 3 322 a b 6 128a b
Coefficients: The number in front of the variable is the coefficient. Multiply coefficients.
3 52x 4x 88x
#
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Dividing Powers: If bases are the same subtract exponents.
15
126 6
15 126 36
Negative Exponent: To get rid of a negative exponent flip it over!
4 2 412
116
Zero Exponent: Anything to the zero power is always one!
0 329 1
#