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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Consider each of the following quotients. Each number can be expressed as a power of 3.
81
35
27
34
33
= 32
= 9
= 31
= 33
= 3
27
3
32
31
33
Introduction27
243
= 27
9
Once again, look for a pattern in the quotients shown. If you consider only the exponents, you may notice that
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
Now simplify the following: you consider only the exponents, you may notice that
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!
Simplify the following: you consider only the exponents, you may notice that
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.
Study the two ways shown below to simplify you consider only the exponents, you may notice that
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.
Study the two ways shown below to simplify you consider only the exponents, you may notice that
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
This examples suggests the following definition: you consider only the exponents, you may notice that
What does this suggest?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.
Simplify the following: you consider only the exponents, you may notice that
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
Simplify the following: you consider only the exponents, you may notice that
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