section 5.1 product and power rules for exponents
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Section 5.1
Product and Power Rules for Exponents
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5.1 Lecture Guide: Product and Power Rules for Exponents
Objective 1: Convert between exponential form and expanded form.
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Algebraically
For any natural number n,
with base b and exponent n.
Verbally
For any natural number n, is the product of b used as a ____________ n times. The expression is read as “b to the nth power.”
Numerical Example
Exponential Notation
nb
factors of
n
n b
b b b b
35 5 5 5
24 4 4
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Write each expression in exponential form.
1. 2.
3. 4.
x x x y y 2 x x x x
x x x y y y x y x y x y
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Write each exponential expression in expanded form.
5. 6.3( 5 )a35a
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Write each exponential expression in expanded form.
7. 8.3 2a b 27
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Write each exponential expression in expanded form.
9. 10. 2a b3 2a b
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11. Complete the warm-up examples below:
Expanded Form:
2 5
__________________
______
x x x x x x x x x
Alternate Form:
2 5 2 5
______
x x x
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Algebraically
For any real number x and natural numbers m and n,
Verbally
To multiply two factors with the same base, use the common base and ____________ the exponents
Algebraic Example
_________________
Product Rule for Exponents
.m n m nx x x
2 4x x
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12. 13.
Simplify Each Expression
4 3m m 10 12b b
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14. 15.
Simplify Each Expression
4 6 23r r r 3 52 3x x
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16.
Expanded Form:
Alternate form:
Objective 3: Use the power rule for exponents.
Complete the warm-up examples below:
32 2 2 2
__________________
______
x x x x
32 2 3
______
x x
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Algebraically
For any real number x and natural numbers m and n,
Verbally
To raise a power to a power, ____________ the exponents
Algebraic Example
_________________
Power Rule for Exponents
( ) .m n mnx x
2 5( )x
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17. 18.
Simplify Each Expression
43x 32x
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19. 20.
Simplify Each Expression
23x 23x
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21.
Expanded Form:
Shortcut:
Complete the warm-up examples below:
3
_________
xy xy xy xy
xxx yyy
3_________xy
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Algebraically
For any real numbers x and y and any natural number, m,
Verbally
To raise a product to a power, raise each ___________ to this power.
To raise a quotient to a power, raise both the ________ and the _________ to this power.
Algebraic Example
_________________
_________________
Raising Products and Quotients to a Power
,m m mxy x y
for 0.m m
m
x xy
y y
5xy
5x
y
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22. 23.
Simplify Each Expression
3mn 2 4( )x y
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24. 25.
Simplify Each Expression52
3
mn
3xy
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26. 27.
Simplify Each Expression
42x 4
2x
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28. 29.
Simplify Each Expression
2
3
7x
23
4
35ab
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30. Evaluate the expression for 2 4b ac 4,a 2,b 3.c and
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Comparing Addition and Multiplication
Add the like terms and simplify the products.
31. 32.2 2 2x x x 2 2 2x x x
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33. 34.4 43 5x x 4 43 5x x
Comparing Addition and Multiplication
Add the like terms and simplify the products.
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35. 36.If possible, add If possible, multiply 2 32 5x x 2 32 5x x
Comparing Addition and Multiplication
Add the like terms and simplify the products.
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37. The formula for the area of a square is where x is the length of a side of the square.
(a) Complete the table of values for the area of a square with sides of length x cm.
Length of a side, x cm Area, cm2
1
2
3
4
2A x
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37. The formula for the area of a square is
(b) How does the area of a square with sides of length 2 cm compare to the area of a square with sides of length 4 cm?
2 cm
4 cm
2A xwhere x is the length of a side of the square.