Simplifying Exponential EquationsDate: 08/19/2003 at 17:51:41 From: Mark Subject: Simplifying exponential equations I have always had trouble with exponents, and this is a review question that I have no clue how to even start. It is (3r^-1s^2t^3) ^2/27r^3st^-2. I don't even know how to begin! Can you help? Date: 08/20/2003 at 13:46:30 From: Doctor Ian Subject: Re: Simplifying exponential equations Hi Mark, You can find a simple explanation of the important rules here: Properties of Exponents http://mathforum.org/library/drmath/view/57293.html Be sure to read it before continuing. Given your example, (3 r^-1 s^2 t^3)^2 ------------------ 27 r^3 s t^-2 the easiest rule to apply would be (ab)^c = (a)^c (b)^c We can use this to rewrite the numerator: (3)^2 (r^-1)^2 (s^2)^2 (t^3)^2 ------------------------------- 27 r^3 s t^-2 Now the easiest rule to apply is (a^b)^c = a^(bc) That gives us 9 r^(-2) s^4 t^6 --------------------- 27 r^3 s t^-2 Now, at this point, it helps to keep things straight if we 'unmultiply' the fraction: 9 r^(-2) s^4 t^6 -- * ------- * --- * ---- 27 r^3 s t^-2 Why would we do this? Because now we can apply the rule a^b --- = a^(b-c) a^c For example, t^6 ---- = t^(6 - -2) = t^(6 + 2) = t^8 t^-2 So now we have 9 r^(-2) s^4 -- * ------- * --- * t^8 27 r^3 s I'll leave the rest to you, but write back if you're still stuck. As for the logic of these problems, it's basically the same as the logic of most of mathematics: You start out with something that doesn't look the way you want it to, and you apply successive transformations to it. If each transformation is guaranteed to preserve the meaning of the thing, then you can change appearance without changing meaning. For example, this is all you're really doing when you evaluate an arithmetic expression, 3 * (4 - 2) + 6 / 3 = 3 * 2 + 6 / 3 Because 4 - 2 = 2 = 6 + 6 / 3 Because 3 * 2 = 6 = 6 + 2 Because 6 / 3 = 2 = 8 Because 6 + 2 = 8 or when you substitute a value into an equation, y = 3x - 4 y = 3*2 - 4 Legal if we know that x = 2 y = 6 - 4 Because 3 * 2 = 6 y = 2 Because 6 - 4 = 2 or when you solve a formula for one variable in terms of others, V = pi r^2 h Volume of cylinder V/pi = r^2 h Dividing both sides by the same (nonzero) quantity preserves equality. V/pi/h = r^2 Same thing. _______ ____ \| V/pi/h = \| r^2 Taking the square root of both sides preserves equality. _______ ____________ \| V/pi/h = r Because \| something^2 = something or when you integrate, or differentiate, or prove a trig identity, and so on, and so on. It's really just the same game, played in slightly different contexts, i.e., you have rules that say "Whenever I see an instance of THIS, I can replace it with an instance of THAT," and then you search for opportunities to apply these rules. How do you know which rules to apply at any given time? That's sort of like knowing which piece to move next in a game of chess. Largely it's a case of knowing what to do in a particular situation because you've been in that situation before. I hope this helps! - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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