6.3 Dividing Monomials. CORD Math Mrs. Spitz Fall 2006. Okay, for the HW. Scale: How many correct? 1720 – 20 points—not bad – you have it! 1216 – 15 points – You need some practice 711 – 10 points. You need some help. Practice some more – rework the problems missed
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81
35
27
34
33
= 32
= 9
= 31
= 33
= 3
27
3
32
31
33
Introduction27
243
= 27
9
4 – 3 = 1, 3 – 1 = 2, and 5 – 2 = 3
81
35
27
34
33
= 32
= 9
= 31
= 33
= 3
27
3
32
31
33
Introduction27
243
= 27
9
b5
=
b2
Quotient of PowersQuotient of Powers:
For all integers m and n, and any nonzero number a,
b ≠ 0
am
= amn
an
b · b · b · b · b
= b · b · b
b · b
= b3
These examples suggest that to divide powers with the same base, you can subtract the exponents!
a4
b3
a1
b2
Example 1a4b3
=
ab2
= a41b32
Group the powers that have the same base.
= a3b1
Subtract the exponents by the quotient of powers property.
= a3b
Recall that b1 = b.
a3
a3
Next note:a3
a3
a · a · a
=
a33
=
a3
a · a · a
a3
= a0
= 1
Zero Exponent:
For any nonzero number a, a0 = 1.
k2
k2
k7
k7
Aha:k2
k · k
k2
=
k27
=
k7
k7
k · k · k · k · k · k · k
= k5
1
=
k · k · k · k · k
Since cannot have two
different values, we can conclude that k5
1
=
k5
1
=
k5
Negative Exponents:
For any nonzero number a and any integer n, an
1
=
an
To simplify an expression involving monomials, write an equivalent expression that has positive exponents and no powers of powers. Also, each base should appear only once and all fractions should be in simplest form.
s5
1
6
r3
1
1
1
t2
3
r7
s5
t2
3
18
Example 26r3s5
=
·
·
·
18r7s5t2
= r3(7)s55t2
Recall = t2
= r10s0t2
Subtract the exponents.
= r10t2
Remember that s0 = 1.

3
42
a2
=
·
22
a8
1
1
= 64a6
= 43a6
Example 3(4a1)2
Power of a product property
(2a4)2
42
a2
=
Simplify
4a8
= 421a28
Subtract the exponents
= 43a6
Definition of negative exponents
Simplify