Question: Powers of 2

Comment on Powers of 2

Can you also solve this question without factoring? If yes, could you explain how?
Kind regards, Lisa
gmat-admin's picture

Hi Lisa,
Sure, here's an approach that doesn't require factoring:
First evaluate the NUMERATOR:
2^(-13) - 2^(-9) = 1/(2^13) - 1/(2^9)
= 1/(2^13) - (2^4)/(2^13) [found common denominator]
= (1 - 2^4)/(2^13)

First evaluate the DENOMINATOR:
2^(-13) - 2^(-17) = 1/(2^13) - 1/(2^17)
= (2^4)/(2^17) - 1/(2^17) [found common denominator]
= (2^4 - 1)/(2^17)

So, NUMERATOR/DENOMINATOR = [(1 - 2^4)/(2^13) ]/[(2^4 - 1)/(2^17)]
= [(1 - 2^4)(2^17)]/[(2^4 - 1)/(2^13)]
Now recognize that (1 - 2^4)/(2^4 - 1) = -1
So, our fraction = -[(2^17)/(2^13)]
= -(2^4)
= -16

Hi Brent,
Thank you for the help here.
In my shortcut search as usual, I followed your idea of "something" again.
I noticed that the numerator would be a negative "SOMETHING" while the denominator would be positive "something" so the answer has to be "something" that is negative and greater than 1. Only A fit.
gmat-admin's picture

Perfect reasoning! Great shortcut!!

Hi Brent, I am lost about something that could be negative and greater than 1.
Could you guide me where I am wrong?
gmat-admin's picture

Great question, Esguitar!

I should have clarified abrahamic01's original comment so as not to confuse others.

I believe abrahamic01 meant so say that the MAGNITUDE (aka absolute value) of the correct answer is greater than 1.

In that sense, -16 has a magnitude that's greater than 1

Does that help?


I just want to point out that once both the numerator and denominator is factored, there is no need to actually calculate (1 - 2^4) / (2^4 - 1) under the general rule that (x-y)/(y-x) always = -1, so we can just calculate 2^4 * -1 = -16.
gmat-admin's picture

Good point.
That would save us the step where we got (-15)/15 = -1


Hi Brent,

This may be a basic question but why do you factor 2^-13 and 2^-17 instead of 2^-9 and 2^-13? Do you factor the lowest number? I feel like I am missing some of the basic rules here.
gmat-admin's picture

Hi Aaron,

Your instincts are correct;we factor out those powers because they're the SMALLEST values.

Let's check out some examples that don't involve negative exponents:
k^5 - k^3 = k^3(k^2 - 1)
m^19 - m^15 = m^15(m^4 - 1)
y^4 + y^3 - y^8 = y^3(y + 1 - y^5)
Notice that, each time, the greatest common factor of the terms is the term with the SMALLEST exponent.

So, in the expression 2^(-13) - 2^(-17), the term with the SMALLER exponent is 2^(-17), since -17 < -13
So, we factor out 2^(-17)

Here are a few additional examples (for reinforcement):
w^x + x^(x+5) = w^x(1 + w^5)
2^x - 2^(x-2) = 2^(x-2)[2^2 - 1]

Does that help?

Can you flip the numerator and denominator before solving so dealing with positive exponents instead of negative? Easier to think about factoring if are positive and believe this results in the same solution.
gmat-admin's picture

Yes you can flip the fraction, simplify it, and then flip the final result to get your answer, but I don't think that'll be any easier.
Can you show me what you mean?

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