I love the question and rerika's attempt to post the exponents as actual exponents (rather than the x^y) form, but it didn't quite translate.
I think I can break down the question, though (and it fits with the answer choices). It should read:
If 3^x - 3^(x-1) = 162, then what is x(x-1)?
A) 12
B) 16
C) 20
D) 30
E) 81
As it's written, the answer is C.
Explanation (I'll put in "spoiler" text since many have not been able to read the question yet):
[spoiler]It may seem awkward to do, but you can factor out a 3^(x-1) from both terms on the left hand side. 3^x can be written as 3^1 * 3^(x-1) in order to create that duplicate term to factor:
3^x - 3^(x-1) = 162
3^1 * 3^(x-1) - 3^(x-1) = 162
[3^(x-1)] (3-1) = 162
[3^(x-1)] * 2 = 162
Then you can divide both sides by 2 to get:
3^(x-1) = 81
As with any exponent problem with variables in the exponents, you should try to create the same bases:
3^(x-1) = 3^4
x - 1 = 4
x = 5
x(x-1) = 5*4 = 20
That's one, fairly direct way to do it. Another way would be to recognize that you need to find 3s on the right hand side of the equation to be able to do anything with the 3^variable terms on the left, and factor 162 into 2*3^4. Then you could either backsolve combinations of x and x-1 or work to find patterns with 3s and consecutive exponents.
Ultimately, I'd say that the biggest takeaway on this problem is that, when exponents appear with variables in the exponents, you should try to find common bases, and usually to do that you'll want to break numbers down into prime factors. [/spoiler]
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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