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## Comment on

Powers of 2## Can you also solve this

Kind regards, Lisa

## Hi Lisa,

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.

## Perfect reasoning! Great

Perfect reasoning! Great shortcut!!

## Hi Brent, I am lost about

Could you guide me where I am wrong?

## Great question, Esguitar!

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.

## Good point.

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.

## Hi Aaron,

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

## Yes you can flip the fraction

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?