6.3 Dividing Monomials6.3 Dividing MonomialsCORD MathMrs. SpitzFall 2006

Okay, for the HWOkay, for the HW

Scale: How many correct? 17-20 – 20 points—not bad – you have it! 12-16 – 15 points – You need some practice 7-11 – 10 points. You need some help. Practice

some more – rework the problems missed 6 and below – you need some significant help

in order to complete this. Take the worksheet and have mom or dad sign it. Rework problems

Turn it in for credit in the box! Record your scores

Quiz after 6.3 is graded next time we meet!

Standard/ObjectiveStandard/Objective

Standard: Students will understand algebraic concepts and applications

Objectives: Students will simplify expressions

involving quotients of monomials, andSimplify expressions containing negative

exponents

AssignmentAssignment

WS 6.3Quiz – end of the 6.2 – 20 minutesMid-chapter Test after 6.4Quiz after 6.6Test after 6.9 – short answer – show

all work

IntroductionIntroduction

Consider each of the following quotients. Each number can be expressed as a power of 3.

8127

= 327

273

= 9

2439

= 27

34

33= 31

33

31= 32

35

32= 33

IntroductionIntroduction

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

8127

= 327

273

= 9

2439

= 27

34

33= 31

33

31= 32

35

32= 33

Quotient of PowersQuotient of Powers

Now simplify the following:

b5

b2= b ≠ 0

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!

Quotient of Powers:

For all integers m and n, and any nonzero number a,

am

an= am-n

Example 1Example 1

Simplify the following:

a4b3=

ab2

a4

a1

b3

b2

= a4-1b3-2

= a3b1

= a3b

Group the powers that have the same base.

Subtract the exponents by the quotient of powers property.

Recall that b1 = b.

Next note:Next note:

Study the two ways shown below to simplify

a3=

a3

a · a · a

= 1

a · a · a

a3

a3

a3=

a3a3-3

= a0

Zero Exponent:

For any nonzero number a, a0 = 1.

Aha:Aha:

Study the two ways shown below to simplify

k2=

k7 k · k · k · k · k · k · k

k · k

k2

k7

k2=

k7k2-7

= k-5

=k · k · k · k · k

1

=k5

1

k2

k7Since cannot have two

different values, we can conclude that k-5 =

k5

1

What does this suggest?What does this suggest?

This examples suggests the following definition:

Negative Exponents:

For any nonzero number a and any integer n, a-n =an

1

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.

Example 2Example 2

Simplify the following:

-6r3s5=

18r-7s5t-2

-618

r3

r-7

= r3-(-7)s5-5t2 Recall = t2

· 1t-2

s5

s5··

-1

31t-2

= r10s0t2-1

3

= r10t2

3-

Subtract the exponents.

Remember that s0 = 1.

Example 3Example 3 Simplify the

following:

(4a-1)-2

(2a4)2Power of a product property= 4-2

22

a2

a8·

= 4-2

4a8

a2

= 4-2-1a2-8

= 4-3a-6

= 43a61

= 64a61

Simplify

Subtract the exponents

Definition of negative exponents

Simplify