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?

Cheers,
Brent

Hello,
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

Cheers,
Brent

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?

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