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Expanded Form:Shortcut: 1. Complete the warm-up examples below. Assume (in exponential form)
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Section 5.2
Quotient Rule and Zero Exponents
5.2 Lecture Guide: Quotient Rule and Zero Exponents
Objective: Use the quotient rule for exponents.
Expanded Form: Shortcut:
1. Complete the warm-up examples below.
Assume 0x
5
2
1______
x x x x x xx x x
x x x
(in exponential form)
55 2
2
______
x xx
Algebraic Examples
Algebraically For any nonzero real number x and natural numbers m and n with m > n,
Verbally To divide two expressions with the same base, use the common base and ____________ the exponents.
____________
Quotient Rule for Exponents where m > n:
mm n
n
x xx
6
2
xx
m
n
xx
for 0x
is undefined for 0x
Simplify each expression.
2.6
2
dd
3.6 6
4 2
m nm n
Simplify each expression.
4.8
6
77
Simplify each expression.
5.
Simplify each expression.
10
426xx
6.
Simplify each expression.
20
102010xx
7.
Simplify each expression.
8 14
2 4
122x yx y
Expanded Form: Shortcut:
8. Complete the warm-up examples below. Assume 0x
22 5
5
______
x xx
2
5
1
______
x x xx x x x x x
x x x
Quotient Rule for m<n
The expression m
n
xx
can also be simplified when m < n.
The reasoning is similar to that when m > n.
If 0x and m and n are natural numbers, then 1m
n n m
xx x for m < n
In the next section, we will see that two rules are not necessary.
Simplify each expression.
9.2
8
mm
10.4 2
12 6
x yx y
Simplify each expression.
11.
Simplify each expression.
3 8
9 26
12a ba b
12.
Simplify each expression.
5 4
10 82550x yx y
13.
Expanded Form: Shortcut:
Objective: Simplify expressions with zero exponents.
Complete the warm-up examples below.
3
3
3 3
______
x x x xx x x x
x
(in exponential form)
3
3 _____xx
(as a simplified fraction, because any nonzero number divided by itself = ______)
Assume 0x
Algebraic Example
For any nonzero real number x,
Any nonzero real number raised to the 0 power equals 1.
____________
Zero Exponents
0 1x
is undefined00
012
Algebraically
Verbally
Simplify each expression, assuming all bases are nonzero.
14. 06a
15. 06a
Simplify each expression, assuming all bases are nonzero.
16. 06a b
Simplify each expression, assuming all bases are nonzero.
17. 0 06a b
Simplify each expression, assuming all bases are nonzero.
18.
Simplify each expression if possible.
(a)
(d)(c)
(b)0 3 3 0
30 03
Objective: Combine the properties of exponents to simplify expressions.
Summary of the Exponent Rules:For any nonzero real numbers x and y and whole number exponents m and n,
Product rule: m nx x _________
Power rule:
( ) _________ ( ) _________ _________m
m n m xx xyy
Quotient rule: m
n
xx
m
n
xx
____________ for m > n
____________ for m < n
Zero exponent: 0x ____________ for 0x
Simplify each expression. Assume all bases are nonzero.
19.45
32xx
Simplify each expression. Assume all bases are nonzero.
20.8 10
12 6
2821x yx y
Simplify each expression. Assume all bases are nonzero.
21. 22 43 5x x
Simplify each expression. Assume all bases are nonzero.
22.22 8
6 4248x yx y
Simplify each expression. Assume all bases are nonzero.
23.3 7
2 436 1512 45x xx x
Simplify each expression. Assume all bases are nonzero.
24.
23 5
43
5
3
x y
xy
25. Use the formula (1 )tA P r to find the value ofa $15,000 investment compounded at 9% at the end of each year over a six-year period.
Year, t Value, A
0 $15,000
1
2
3
4
5
6
Comparing Subtraction and DivisionSubtract the like terms and simplify the quotients. Assume 0x
26. 3 312 4x x
Comparing Subtraction and DivisionSubtract the like terms and simplify the quotients. Assume 0x
27. 3
3124xx
Comparing Subtraction and DivisionSubtract the like terms and simplify the quotients. Assume 0x
28. If possible, subtract 316 4x x
Comparing Subtraction and DivisionSubtract the like terms and simplify the quotients. Assume 0x
29. If possible, divide 316
4xx
