Exponent Challenge Question Series: Day 1

by on September 26th, 2010

As evidenced by the frequency in which they occur on the Beat The GMAT forums, exponent-based algebra problems can rank among the hardest questions on the GMAT.  The authors of these questions know that simple rules ( x^{2} * x^{3} = x^{(2+3)}, for example) can be made exponentially harder if multiple bases are used, variables appear in the exponents themselves, and large numbers enter the equation.

Over the next five days on Beat The GMAT, Veritas Prep’s authors will each day contribute a difficult exponent-related challenge problem, with a solution to follow the next day. Before you begin, you may want to consider this as a way to crack the exponent code; nearly all exponent-based problems can be solved using a combination of these three guiding exponent principles:

  1. Exponent rules are almost all related to multiplication and division with virtually no rules that directly apply to addition and subtraction.  When facing exponent problems, look for opportunities to factor and multiply to put yourself in a position to use your multiplication-heavy exponent expertise.
  2. Most exponent rules require you to have common bases in order to apply them, so look to break down bases into prime factors so that you have common bases with which to work.
  3. Exponents are very pattern-driven, so when large numbers are present you can often establish a rule using small numbers and then extrapolate it to the larger ones.

In the problems that follow over the next five days, you’ll have opportunities to put these strategies into practice as you apply your exponential expertise!

Question 1

If 8^{x}9 ^{2y} = 81(2^{12y}), what is the value of x?

(A)   2
(B)   4
(C)   8
(D)   12
(E)    16


  • breaking down bases in terms of prime factors
    Given : 8^x . 9^2y = 81(2^12y)
    to find x ?
    8^x . 9^2y = 81(2^12y)
    => ((2)^3)^x . ((3)^2)2y = (3)^4(2^12y)
    : Using the rule (a)^x)^y = a^xy we multiply the powers
    => 2^3x . 3^4y = 3^4 . 2^12y

    it is sufficient to deduce that
    1. 3x = 12y
    2. 4y = 4

    from 2 we get y =1
    substituting value of y in equation 1 we get x = 4.
    Hence the correct option is (B).

  • Answer B

  • y=1 and x=4

    So Answer B

  • The answer is B,
    y = 1 and x = 4...Jitendra does a pretty decent job at explaining why :)

    • B

  • The Answer is B (4)

  • answer is B -- 4

  • I must admit that I never thought I would get to a point where I could do problems like this one in 15 seconds.

    • You've got it right where you want it, Richard - now finish if off!

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