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Multiplying Monomials (7-1) Objective: Multiply monomials. Simplify expressions involving monomials. Monomials A monomial is a number, a variable, or the product of a number and one or more variables with nonegative integer exponents. It has only one term. An expression that involves division by a variable is not a monomial. The monomial 3x is an example of a linear expression since the exponent of x is 1. The monomial 2x2 is a nonlinear expression since the exponent is a positive number other than 1. Example 1 Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. a. 17 – c b. 8f2g c. Yes, this is a product of numbers and variables. ¾ d. No, this expression has two terms. Yes, this is a constant. 5/ t No, there is a variable in the denominator. Check Your Progress Choose the best answer for the following. Which expression is a monomial? A. x5 B. 3p – 1 C. 9x/y D. c/d Powers Recall that an expression of the form xn is called a power and represents the result of multiplying x by itself n times. x is the base, and n is the exponent. The word power is also used sometimes to refer to the exponent. exponent 34 = 3 ∙ 3 ∙ 3 ∙ 3 = 81 base Product of Powers By applying the definition of a power, you can find the product of powers. Look for a pattern in the exponents. 22 ∙ 24 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 26 43 ∙ 42 = 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 = 45 To multiply two powers that have the same base, add their exponents. For any real number a and any integers m and p, am ∙ ap = am+p. b3 ∙ b5 = b3+5 = b8 g4 ∙ g6 = g4+6 = g10 Example 2 Simplify each expression. a. (r4)(-12r7) = = b. (1 ∙ -12)(r4 ∙ r7) -12r11 (6cd5)(5c5d2) = = (6 ∙ 5)(c1 ∙ c5)(d5 ∙ d2) 30c6d7 Check Your Progress Choose the best answer for the following. A. Simplify (5x2)(4x3). A. B. C. D. 9x5 20x5 20x6 9x6 (5 ∙ 4)(x2 ∙ x3) Check Your Progress Choose the best answer for the following. B. Simplify 3xy2(-2x2y3). A. B. C. D. 6xy5 -6x2y6 1x3y5 -6x3y5 (3 ∙ -2)(x1 ∙ x2)(y2 ∙ y3) Power of a Power We can use the Product of Powers Property to find the power of a power. In the following examples, look for a pattern in the exponents. (32)4 = (32)(32)(32)(32) = 32+2+2+2 = 38 (r4)3 = (r4)(r4)(r4) = r4+4+4 = r12 To find the power of a power, multiply the exponents. For any real number a and any integers m and p, (am)p = a m∙p. (b3)5 = b3∙5 = b15 (g6)7 = g6∙7 = g42 Example 3 Simplify [(23)3]2. = = = 23∙3∙2 218 262,144 Check Your Progress Choose the best answer for the following. Simplify [(42)2]3. A. B. C. D. 47 48 412 410 42∙2∙3 Power of a Product We can use the Product of Powers Property and the Power of a Power Property to find the power of a product. In the following examples, look for a pattern in the exponents. (tw)3 = (tw)(tw)(tw) = (t1 ∙ t1 ∙ t1)(w1 ∙ w1 ∙ w1) = t3w3 (2yz2)3 = (2yz2)(2yz2)(2yz2) = (2)3(y)3(z2)3 = 8y3z6 To find the power of a product, find the power of each factor and multiply. For any real numbers a and b and any integer m, (ab)m = ambm. (-2xy3)5 = (-2)5(x)5(y3)5 = -32x5y15 Example 4 Express the volume of a cube with side length 5xyz as a monomial. V = s3 V = (5xyz)3 V = 53x3y3z3 V = 125x3y3z3 Check Your Progress Choose the best answer for the following. Express the surface area of the cube as a monomial. A. B. C. D. 8p3q3 24p2q2 6p2q2 8p2q2 SA = 6s2 SA = 6(2pq)2 SA = 6 ∙ 22p2q2 SA = 6 ∙ 4p2q2 Simplify Expressions We can combine and use these properties to simplify expressions involving monomials. To simplify a monomial expression, write an equivalent expression in which: Each variable base appears exactly once. There are no powers of powers, and All fractions are in simplest form. Example 5 Simplify [(8g3h4)2]2(2gh5)4. = = = = (8g3h4)4(2gh5)4 84(g3)4(h4)4 ∙ 24(g)4(h5)4 (4096 ∙ 16)(g12 ∙ g4)(h16 ∙ h20) 65,536g16h36 Check Your Progress Choose the best answer for the following. Simplify [(2c2d3)2]3(3c5d2)3. A. B. C. D. 1728c27d24 6c7d5 24c13d10 5c7d21 = (2c2d3)6 ∙ (3c5d2)3 = 26(c2)6(d3)6 ∙ 33(c5)3(d2)3 = (64 ∙ 27)(c12 ∙ c15)(d18 ∙ d6)