If 2^x - 2^(x - 2) = 3*2^13 what is the value of x?

NO idea how to solve this still, particulalrly step 3-4 . Totally confused.

Hi Rags2Riches,

Most Quant questions on the GMAT are written so that they can be approached in more than one way. By extension, if you're reading through an explanation that involves lots of step-heavy techincal math, then there is likely a Tactical approach that would be faster and easier.

Factoring the equation is a useful approach here. You can actually take it a step further by TESTing THE ANSWERS (you might find it easier to manipulate numbers than to manipulate variables). One of those numbers IS the value of X, so you can plug the answers in, do the necessary math and find the one value that balances out the equation. Here's how to approach it:

Since the "right side" of the equation is greater than 2^14, X must be bigger than 14 (as the "left side" of the equation involves subtraction). So we can eliminate A, B and C.

Let's TEST answer D: 15

If X = 15, then...

2^15 - 2^13 can be factored into...
(2^13)(2^2 - 1) =
(2^13)(3)

This is EXACTLY what's on the "right side" of the equation, so X MUST be 15.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at: [email protected]

Hey there - did you happen to check out our video solution? Right here: https://gmatclub.com/forum/if-2-x-2-x-2 ... l#p3150333

Or are you looking for a specific explanation of those specific steps?

Let us know

Although testing options is an important and effective technique for the GMAT, it is still important to have the ability to work with algebra. So, here are steps 3 and 4 in more details:

\(2^{-2}=\frac{1}{2^2}\), thus \(2^x-2^{x}*2^{-2}=3*2^{13}\) becomes \(2^x-\frac{2^{x}}{2^{2}}=3*2^{13}\). Here we can factor out 2^x from \(2^x-\frac{2^{x}}{2^{2}}\) and get \(2^x(1-\frac{1}{2^{2}})=3*2^{13}\).

If you are still unclear about the above,, I recommend checking out the following links.

7. Algebra

• Theory
  Math : Algebra 101
  Algebra: Tips and hints
  FOIL on the GMAT: Simplifying and Expanding
  Factoring Quadratics
  Solving Quadratic Equations
  Using Algebra vs. Logic on GMAT Quant Questions
  
  

  8. Exponents and Roots of Numbers

  
  
  • Theory
    Exponents and Roots on the GMAT: Tips and hints
    Cyclicity and the remainders on the GMAT
    
    • Questions
      Units digits, exponents, remainders problems
      Tough and tricky DS exponents and roots questions with detailed solutions
      Tough and tricky PS exponents and roots questions with detailed solutions
      Exponents DS Questions
      Exponents PS Questions
      Roots DS Questions
      Roots PS Questions

Check below for more:
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Ultimate GMAT Quantitative Megathread

Alternate solution.
Since x is positive x should be more than 14. Since \(2^x- positive number = 3*2^{13}\), where 3>2, so RHS is more than 2^14. Eliminates A,B,C. Can put values to see which one of the 15 or 17 works to avoid working with x

how 2 to the power 15

\((4-1)*2^{13}=\)

\(=(2^2-1)*2^{13}=\)

\(=2^2*2^{13} + 1* 2^{13} =\)

\(=2^{15}-2^{13}\)

Hope it helps.