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ball at timet. The acceleration due to gravity isg= 32 feet per second per second. If the initialvelocity of the ball isv0= 272 feet per second, find the speed of the ball aftert= 6 seconds.48. A ball is thrown vertically upward. Its velocitytseconds after its release is given by the formulav=v0-gt,wherev0is its initial velocity,gis the acceleration due to gravity, andvis the velocity of theball at timet. The acceleration due to gravity isg= 32 feet per second per second. If the initialvelocity of the ball isv0= 470 feet per second, find the speed of the ball aftert= 4 seconds.49.Evennumbers.Evaluatetheex-pression2nforthefollowingvalues:i)n= 1ii)n= 2iii)n= 3iv)n=-4v)n=-5vi)Is the result always an even num-ber? Explain.ii)n= 2iii)n= 3iv)n=-4v)n=-5vi)Is the result always an odd number?Explain.❧❧❧Answers❧❧❧1.-1863.-245.77.139.13811.-13413.115.-7217.-219.121.6923.025.9
186CHAPTER 3.THE FUNDAMENTALS OF ALGEBRA27.3629.-631.-7133.135.537.4639.-2941.256 feet43.110 degrees45.-409◦F47.80 feet per second49.i)2ii)4iii)6iv)-8v)-10vi)Yes, the result will always be aneven number because 2 will alwaysbe a factor of the product 2n.
3.3.SIMPLIFYING ALGEBRAIC EXPRESSIONS1873.3Simplifying Algebraic ExpressionsRecall the commutative and associative properties of multiplication.The Commutative Property of Multiplication.Ifaandbare any inte-gers, thena·b=b·a,or equivalently,ab=ba.The Associative Property of Multiplication.Ifa,b, andcare any inte-gers, then(a·b)·c=a·(b·c),or equivalently,(ab)c=a(bc).The commutative property allows us to change the order of multiplicationwithout affecting the product or answer. The associative property allows us toregroup without affecting the product or answer.You Try It!EXAMPLE 1.Simplify: 2(3x).Simplify:-5(7y)Solution.Use the associative property to regroup, then simplify.2(3x) = (2·3)xRegrouping with the associative property.= 6xSimplify: 2·3 = 6.Answer:-35yThe statement 2(3x) = 6xis anidentity. That is, the left-hand side andright-hand side of 2(3x) = 6xare the samefor all values ofx. Although thederivation inExample 1should be the proof of this statement, it helps theintuition to check the validity of the statement for one or two values ofx.Ifx= 4, then2(3x) = 2(3(4))and6x= 6(4)= 2(12)= 24= 24Ifx=-5, then2(3x) = 2(3(-5))and6x= 6(-5)= 2(-15)=-30=-30The above calculations show that 2(3x) = 6xfor bothx= 4 andx=-5.Indeed, the statement 2(3x) = 6xis true, regardless of what is substituted forx.
188CHAPTER 3.THE FUNDAMENTALS OF ALGEBRAYou Try It!EXAMPLE 2.Simplify: (-3t)(-5).Simplify:(-8a)(5)Solution.In essence, we are multiplying three numbers,-3,t, and-5, butthe grouping symbols ask us to multiply the-3 and thetfirst. The associativeand commutative properties allow us to change the order and regroup.