Fill in the Table and look for any patterns
Simplify:
Rewrite with a
Write answer
using a positive
exponent
negative exponent
Quotient
of
Powers
Expanded Form
a)
52
53
5i 5
5i 5i 5
1
5
( 5)
b)
53
52
5i 5i 5
5i 5
5
=5
1
1
⎛1⎞
or ⎜ ⎟
−1
5
⎝5⎠
c)
x5
x2
d)
x2
x5
e)
x 23
x14
f)
x14
x 23
x 94
x 36
g)
−1
−1
Rule:
MCC@WCCUSD 02/15/12
Quotient of Powers
Standards:
2.0 Understand and use the rules of exponents
10.0 Divide monomials. Solve multi-step problems, including word problems, by using this
technique(Develop)
Objective: Today I will divide powers with the same base and raise a quotient to a power.
Lesson Opener:
Activity can be done as a class or students can fill in the chart in partners/groups.
Debrief:
“Did you notice any patterns?” [Answers will vary]
- same bases
- all had same coefficient
- Do not need to expand each expression. When you divide powers with the same base,
subtract their exponents
- Negative exponent is the reciprocal of the base
to write
using
positive
exponents
- When there are more factors in the numerator than denominator: x n n is positive
1
- When there are more factors in the denominator than numerator: n
x
Quotient of Powers Property: Dividing powers with the same base
For every non-zero number a and integers m and n
am
= a m−n
n
a
where a ≠ 0
Ex 1) Simplify each expression.(write using positive exponents)
a)
c)
44
4
4i4i4i4
=
4
3
=4
7
b)
44
4
= 4 4−1
= 43
z
z3
ziziziziziziz
=
ziziz
4
=z
7
z
z3
= z 7 −3
=z
4
25
26
2i2i2i2i2
=
2i2i2i2i2i2
1
=
2
x4 y2
d)
x2 y3
xixixixi yi y
=
xixi yi yi y
x2
=
y
Page 1 of 4
25
26
= 2 5− 6
= 2 −1
1
=
2
x4 y 2
x2 y3
= x 4−2 y 2−3
= x 2 y −1
x2
=
y
MCC@WCCUSD 2/15/12
Quotient of Powers
You try!
Simplify
a)
x3
x5
xixix
xixixixix
1
= 2
x
=
b)
x3
x5
= x 3−5
= x −2
1
= 2
x
3x 6
x3
3i x i x i x i x i x i x
=
xixix
3
= 3x
3x 6
x3
= 3x 6−3
= 3x3
Ex 2) Simplify
3x 5 i x 2
6x 3
3i x i x i x i x i x i x i x
=
2 i 3i x i x i x
x4
=
2
3x 5 i x 2
6x 3
3x 5+2
=
6x 3
3x 7
=
2 i 3i x 3
x 7−3
=
2
x4
=
2
You Try!
Simplify
a)
4
3
z 4 i10z 3
z2
10z 4+3
=
z2
= 10z 7−2
z i10z
z2
= z 4 i10z 3−2
= 10z 5
= 10z 5
b)
= z 4 i10z1
= 10z 4+1
Page 2 of 4
x i 7x 5
9x 4
7x 5+1
=
9x 4
7x 6
= 4
9x
7x 6−4
=
9
7x 2
=
9
x i 7x 5
9x 4
x i 7x 5−4
=
9
x i 7x
=
9
7x 2
=
9
MCC@WCCUSD 2/15/12
Quotient of Powers
Ex 3) Simplify by using repeated multiplication
a)
⎛ x⎞
⎜⎝ y ⎟⎠
⎛ a⎞
⎜⎝ b ⎟⎠
b)
2
a a a
i i
b b b
aiaia
=
bi bi b
a3
= 3
b
=
x x
= i
y y
=
3
xix
yi y
x2
= 2
y
“ Do you see a pattern? Another way of simplifying without using repeated addition?”
[Answers will vary]
Raising a quotient to a power property:
For every non-zero number a and b and integer n,
Note: Distribution is over addition/subtraction
only. The exponent is NOT being distributed
n
n
⎛a⎞ a
⎜ ⎟ = n , where b ≠ 0
⎝b⎠ b
You try!
Simplify
⎛ −3 ⎞
⎜ ⎟
⎝ y ⎠
a)
⎛ −3 ⎞
⎜⎝ y ⎟⎠
=
=
3
−3( −3) ( −3)
−27
= 3
y
b)
3
−3 −3 −3
i i
y y y
y3
⎛ 2m ⎞
⎜
⎟
⎝ n ⎠
⎛ −3 ⎞
⎜⎝ y ⎟⎠
3
⎛ 2m ⎞
⎜⎝ n ⎟⎠
( −3)
=
3
=
y3
=
=
−3( −3) ( −3)
y
=
16m4
= 4
n
Page 3 of 4
4
2m 2m 2m 2m
i
i
i
n
n
n
n
4
2 ( 2)( 2)( 2) m
3
−27
y3
4
n4
⎛ 2m ⎞
⎜⎝ n ⎟⎠
4
2 4 m4
= 4
n
2 ( 2 ) ( 2 ) ( 2 ) m4
=
n4
16m4
= 4
n
MCC@WCCUSD 2/15/12
Quotient of Powers
Simplify. Write your answer using only positive exponents.
Ex 4)
⎛ 2m ⎞
⎜
⎟
⎝ n ⎠
⎛ 2m ⎞
⎜⎝ n ⎟⎠
⎛ n ⎞
=⎜
⎝ 2m ⎟⎠
−4
−4
4
n
n
n
n
i
i
i
2m 2m 2m 2m
n4
=
2 ( 2 ) ( 2 ) ( 2 ) m4
=
=
n4
16m4
*Remind students that a negative exponent is the reciprocal of the base
⎛ 2m ⎞
⎜⎝ n ⎟⎠
⎛ n ⎞
=⎜
⎝ 2m ⎟⎠
−4
4
n4
= 4 4
2 m
=
n4
2 ( 2 ) ( 2 ) ( 2 ) m4
n4
=
16m4
Page 4 of 4
MCC@WCCUSD 2/15/12