Algebra/Trig Review - Pauls Online Math Notes - Lamar University

Algebra/Trig ReviewBoth of these properties require that at least one of the following is true x ≥ 0 and/ory ≥ 0 . To see why this is the case consider the following example2( )( ) ( i)( i) i4 = 16 = −4 −4 ≠ −4 − 4 = 2 2 = 4 =− 4If we try to use the property when both are negative numbers we get an incorrectanswer. If you don’t know or recall complex numbers you can ignore this example.The property will hold if one is negative and the other is positive, but you can’t haveboth negative.I’ll also need the following property for this problem.n nx = x provided n is oddIn the next example I’ll deal with n even.Now, on to the solution to this example. I’ll first rewrite the stuff under the radical alittle then use both of the properties that I’ve given here.16xy = 8xx yyyy 2y3 6 13 3 3 3 3 3 3 3===8 x x y y y y 2y3 3 3 3 3 3 3 3 3 3 3 3 3 3xxyyyy232xy2 42 32ySo, all that I did was break up everything into terms that are perfect cubes and termsthat weren’t perfect cubes. I then used the property that allowed me to break up aproduct under the radical. Once this was done I simplified each perfect cube and dida little combining.y10.4 8 1516x ySolutionI did not include the restriction that x ≥ 0 and y ≥ 0 in this problem so we’re going tohave to be a little more careful here. To do this problem we will need the followingproperty.n nx = x provided n is evenTo see why the absolute values are required consider 4 . When evaluating this weare really asking what number did we square to get four? The problem is there are infact two answer to this : 2 and -2! When evaluating square roots (or any even rootfor that matter) we want a predicable answer. We don’t want to have to sit down eachand every time and decide whether we want the positive or negative number.Therefore, by putting the absolute value bars on the x we will guarantee that theanswer is always positive and hence predictable.© 2006 Paul Dawkins 10http://tutorial.math.lamar.edu/terms.aspx