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7-1 Multiplying Monomials Algebra 1 Glencoe McGraw-Hill Linda Stamper A monomial is a number, a variable, or a product of a number and one or more variables in which there are no variables in a denominator, 4 x X no variables under a radical sign (√), 4x X and any exponents on the variables are positive integers. x 3 X Monomials that are real numbers are called constants. Determine whether each expression is a monomial. Explain. 4 5 Yes, it is a real number. xy No, it involves addition. x 5 y2 No, it is has a negative power. x3 y 4 Yes, the variables are a product. x y No, it is has a variable in the denominator. a b No, it involves subtraction. Today’s lesson involves simplifying monomials involving powers. To simplify these expressions you will need to memorize the multiplication properties of exponents. Recall an expression like 43 is called a power. base 43 exponent word form: four to the third power four cubed factor form: 4 • 4 • 4 simplified form: 64 Simplify. x3 x2 x x xx x xxxxx How can you get this answer without x5 writing the To multiply powers that have the factored same base, add the exponents. Onform? tonight’s x3 x2 x32 x5 homework, you must show this support work! PRODUCT OF POWERS PROPERTY Simplify. Example 1 Example 2 y 5 y3 a a 3 Example 3 Example 4 3 4 y y y 5 Example 5 4ab6 7 a2b3 x 5 y2 x 7 y Example 6 4ab2c3 6a5b4c2 Simplify. Example 2 Example 1 y 5 y3 y 5 3 a a 3 a 13 a a a a y8 a4 Example 4 Example 3 3 4 5 y y y y 3 4 5 y12 x5 y2 x7 y x57 y21 x12 y3 Simplify. Example 5 4ab6 7 a2b3 4 7 a1 a2 b6 b3 28a12 b63 Example 6 4ab2c3 6a5b4c2 4 6a1 a5 b2 b4 c3 c2 24a15 b2 4 c32 3 9 28a b No multiplication dots in answer! 24a6b6c5 Simplify. y y2 y2 y2 2 3 How can you get this yyyyyy answer 6 y without writing the To find a power of a power, factored multiply the exponents. On form? tonight’s homework, you 3 2 3 must show y2 y this support y6 work! POWER OF A POWER PROPERTY Simplify. Example 8 Example 7 2 2 3 23 2 26 64 m 5 4 m54 m20 Example 9 x 4 3 x2 x 43 x2 x12 x2 x122 x14 POWER OF A PRODUCT PROPERTY To find the power of a product, find the power of each factor and multiply. Simplify. 3xy2 32 x2 y2 9x2 y2 The power is given to each factor inside the parentheses! Simplify. Example 11 Example 10 2w 6 26 w 6 3xy4 34 x4 y 4 81x 4 y 4 6 64w Example 12 Express the area as a monomial. 4ab Example 13 Express the volume as a monomial. 5xyz 4ab Area s2 4 ab 2 42 a2b2 16a2b2 5xyz 5xyz Volume s3 5xyz 3 53 x3y3z3 125x3 y3z3 Using More Than One Property to Simplify. Example 15 Example 14 3x 4 2 3 2 42 3 6 3 3 x 3 x x 3c c5 3 c63 c5 9x8 x3 27c18 c5 9x83 27c185 9x11 27c23 Example 16 2 4 2 53 7 x23y53 2x7 y 7 x y 2xy 3 6 2 6 15 7 7 x y 2x y 3 2 2x67 y157 3 4 13 22 x y 3 Memorize the Properties of Exponents PRODUCT OF POWERS PROPERTY To multiply powers that have the same base, add the exponents. POWER OF A POWER PROPERTY To find a power of a power, multiply the exponents. POWER OF A PRODUCT PROPERTY To find the power of a product, find the power of each factor and multiply. 7-A2 Pages 361-362 # 16–24,29,39–45,48. Algebra rocks!