copyright (c) 2010 pearson education, inc. laws of exponents
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Copyright (c) 2010 Pearson Education, Inc.
Laws of Exponents
Tobey & Slater, Beginning Algebra, 7e 2
An exponent is a “shorthand” number that saves writing the multiplication of the same numbers.
433 3 3 3
exponent34
base
This is read “three to the fourth power.”
Exponents
Tobey & Slater, Beginning Algebra, 7e 3
The Product Rule
Example) Multiply.
The Product RuleTo multiply two exponential expressions that have the same base, keep the base and add the exponents.
xa · xb = xa + b
The Product RuleTo multiply two exponential expressions that have the same base, keep the base and add the exponents.
xa · xb = xa + b
a.) c4 · c5 = c4 + 5 = c9
b.) 3a3 · a6 = 3a3 + 6 = 3a9 c.) 4w2 · 2w5 = (4)(2)w2 + 5 = 8w7
Tobey & Slater, Beginning Algebra, 7e 4
The Quotient Rule
Example:Divide.
The Quotient Rule To divide two exponential expressions that have the same base, keep the base and subtract the exponents. .
The Quotient Rule To divide two exponential expressions that have the same base, keep the base and subtract the exponents. .
Remember that the base does not change.
aa b
bx xx
a.)6
233
6 2 43 3
Tobey & Slater, Beginning Algebra, 7e 5
The Quotient Rule(by cancellation)
Example:Divide.
a.)2
633 4
3 3 13 3 3 3 3 3 3
b.)4
7z
z
31z z z z
z z z z z z z z
Quotient Rule
Tobey & Slater, Beginning Algebra, 7e 6
Negative Exponents
Negative exponents determine POSITION of the expression.Negative exponents DO NOT change the sign of the expression.
Negative Exponent Properties:Negative Exponent Properties:
1nn
xx
1 nn
xx
nm
n m
yxy x
n nyx
y x
Tobey & Slater, Beginning Algebra, 7e 7
Zero Power
The Zero Power Property:
, if x 0 (00 remains undefined).
The Zero Power Property:
, if x 0 (00 remains undefined).0 1x
Example Simplify:3 0
2a ba
1a a a aa a
Tobey & Slater, Beginning Algebra, 7e 8
Raising a Power to a PowerTo raise a power to a power, keep the same base and multiply the exponents.
Raising a Power to a PowerTo raise a power to a power, keep the same base and multiply the exponents.
Raising Exponential Expressions to a Power
ba abx x
Example:Simplify. a.) (x5)3
b.) (y3)3
= x5·3 = x15
= y3·3 = y9
Tobey & Slater, Beginning Algebra, 7e 9
Product Raised to a PowerProduct Raised to a Power
Product Raised to a Power
a a axy x y
c.) (4x3y2)3
Example:Simplify. a.) (2c)3
b.) (5xy)2
= 43x9y6 = 64x9y6
= (2)3c3 = 8c3
= (5)2(xy)2 = 25x2y2
Tobey & Slater, Beginning Algebra, 7e 10
Quotient Raised to a PowerQuotient Raised to a Power
Quotient Raised to a Power
a a
ax xy y
if y 0.
Example:Simplify. a.)
4xy
4
4xy
b.)43
23ab
44 312
4 82
3 81a abb