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Dividing Monomials November 4, 2011 Dividing Monomials Objective To simplify quotients of monomials and to find the greatest common factor (GCF) of several monomials. Dividing Monomials There are three basic rules used to simplify fractions whose denominators and numerators are monomials. The property of quotients allows you to express a fraction as a product. Property of Quotients If a, b, c, and d are real numbers with π β 0 and π β 0, then ππ π π = β ππ π π Example 1 15 3 β 5 3 5 5 5 = = β =1β = 21 3 β 7 3 7 7 7 Simplifying Fractions You obtain the following rule for simplifying fractions if you let π = π inβ¨the property of quotients. If b, c, and d are real numbers with π 0 and π 0, then ππ π = ππ π Simplifying Fractions The following rule allows you to divide the numerator and the denominator of a fractionβ¨by the same nonzero number. In the examples of this lesson, assume that no denominator equals zero. Example 2 Simplify 35 42 solution 35 5 β 7 5 7 5 = = β = 42 6 β 7 6 7 6 Example 3 Simplify π7 π5 Simplify π3 π8 solution solution π7 π5 β π2 2 = π = π5 π5 1 π3 π3 = = π8 π3 β π5 π5 Rule of Exponents for Division Remember: when you multiply powers, you add the exponents. The results of Example 3 show that when you divide powers with the same base, you can subtract the smaller exponent from the greater, if they are different. Rule of Exponents for Division If a is a nonzero real number and m and n are positive integers, then If π > π: If π < π: If π = π: ππ πβπ = π ππ ππ 1 = πβπ π π π ππ =1 π π Example 4 Simplify π₯9 π₯5 Simplify π₯2 π₯7 solution solution π₯9 9β5 4 = π₯ = π₯ π₯5 1 π₯2 1 = 7β2 = 5 7 π₯ π₯ π₯ The Greatest Common Factor The greatest common factor (GCF) of two or more monomials is the common factor with the greatest coefficient and the greatest degree in each variable. Example 5 Find the GCF of 72x3yz3 and 120x2z5 solution Step 1 Find the GCF of the numerical coefficients. 72 = 23 β 32 and 120 = 23 β 3 β 5 οthe GCF of 72 and 120 is 23ο3=8ο3=24. Example 5 Find the GCF of 72x3yz3 and 120x2z5 Solution Step 2 Find the smaller power of each variable that is a factor of both monomials. The smaller power of x is x2 y is not a common factor. The smaller power of z is z3 Example 5 Find the GCF of 72x3yz3 and 120x2z5 solution Step 3 Find the product of the GCF of the numerical coefficients and the smaller power of each variable that is a factor of both monomials. 24 · x2 · z3 ο the GCF of 72x3yz3 and 120x2z5 is 24 · x2 · z3 . Simplified Quotient of Monomials A quotient of monomials is said to be simplified when each base appears only once, when there are no powers of powers, and when the numerator andβ¨denominator have no common factors other than 1. Example 6 Simplify 35π₯ 3 π¦π§ 6 56π₯ 5 π¦π§ Solution 1 Use the property of quotients and the rule of exponents for division. 35π₯ 3 π¦π§ 6 35 π₯ 3 π¦ π§ 6 = β 5β β 5 56π₯ π¦π§ 56 π₯ π¦ π§ 5 5 1 5π§ = β 2 β 1 β π§5 = 8 π₯ 8π₯ 2 Example 6 Solution 2 Find the GCF of the numerator and denominator and use the rule for simplifying fractions. 35π₯ 3 π¦π§ 6 7π₯ 3 π¦π§ β 5π§ 5 5π§ 5 = 3 = 2 2 5 8π₯ 56π₯ π¦π§ 7π₯ π¦π§ β 8π₯ Class work Homework P 191 Oral Exercises: P 192: 1-57 odd 1-36 P 193: Mixed Review