6.3 Dividing Monomials CORD Math Mrs. Spitz Fall 2006

Okay, for the HW Scale: How many correct? – 20 points—not bad – you have it! – 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/Objective Standard: Students will understand algebraic concepts and applications Objectives: Students will simplify expressions involving quotients of monomials, and Simplify expressions containing negative exponents

Assignment WS 6.3 Quiz – end of the 6.2 – 20 minutes Mid-chapter Test after 6.4 Quiz after 6.6 Test after 6.9 – short answer – show all work

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

Introduction 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 3

Quotient of Powers Now simplify the following: b5b5 b2b2 = b ≠ 0 b · b · b · b · b b · b = b · b · b = b 3 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, amam anan = a m-n

Example 1 Simplify the following: a4b3a4b3 = ab 2 a4a4 a1a1 b3b3 b2b2 = a 4-1 b 3-2 = a 3 b 1 = a 3 b Group the powers that have the same base. Subtract the exponents by the quotient of powers property. Recall that b 1 = b.

Next note: Study the two ways shown below to simplify a3a3 = a3a3 a · a · a = 1 a · a · a a3a3 a3a3 a3a3 = a3a3 a 3-3 = a 0 Zero Exponent: For any nonzero number a, a 0 = 1.

Aha: Study the two ways shown below to simplify k2k2 = k7k7 k · k · k · k · k · k · k k · k k2k2 k7k7 k2k2 = k7k7 k 2-7 = k -5 = k · k · k · k · k 1 = k5k5 1 k2k2 k7k7 Since cannot have two different values, we can conclude that k -5 = k5k5 1

What does this suggest? This examples suggests the following definition: Negative Exponents: For any nonzero number a and any integer n, a -n = anan 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 2 Simplify the following: -6r 3 s 5 = 18r -7 s 5 t r3r3 r -7 = r 3-(-7) s 5-5 t 2 Recall = t 2 · 1 t -2 s5s5 s5s5 ·· 3 1 t -2 = r 10 s 0 t 2 3 = r 10 t Subtract the exponents. Remember that s 0 = 1.

Example 3 Simplify the following: (4a -1)-2 (2a 4 ) 2 Power of a product property = a2a2 a8a8 · = 4a 8 a2a2 = a 2-8 = 4 -3 a -6 = 4 3 a 6 1 = 64a 6 1 Simplify Subtract the exponents Definition of negative exponents Simplify