Question: Powers of 2

You could.
(2^-13 - 2^-9) = 2^-9(2^-4 - 1)
(2^-13 - 2^-17) = 2^-13(1 - 2^-4)
Divide to get (2^4)(-1)
Evaluate: -16

Let's start with some straightforward factoring like these examples:
k^5 - k^3 = k^3(k^2 - 1)
m^19 - m^15 = m^15(m^4 - 1)

IMPORTANT: Notice that, each time, the greatest common factor of both terms is the term with the SMALLER exponent.

Let's factor the numerator: 2^(-13) - 2^(-9)
The two exponents are -13 and -9
Since -13 is the smaller exponent, we'll factor out 2^(-13)

So, we get: 2^(-13) - 2^(-9) = [2^(-13)][something goes here]

Now we need to determine what goes in the brackets.The first part is easy...

2^(-13) - 2^(-9) = [2^(-13)][1 - something]

What is "something" going to be?

We know that [2^(-13)] x [something] = 2^(-9)

When we multiply powers with the same bases, we ADD the exponents. So, we need -13 + ?? = -9

Well, -13 + 4 = -9, so the missing exponent is 4

So, we get: 2^(-13) - 2^(-9) = [2^(-13)][1 - 2^4]
That's the numerator.

We can use the same logic with the denominator to determine that...
2^(-13) - 2^(-17) = [2^(-17)][2^4 - 1]

So, our fraction can be rewritten as: [2^(-13)][1 - 2^4]/[2^(-17)][2^4 - 1]

Now we'll simplify this fraction in PARTS

First, [2^(-13)]/[2^(-17)] = 2^4 (we use the quotient law and subtract -17 from -13 to get the exponent 4)

Next, we need to recognize that [1 - 2^4]/[2^4 - 1] = -1

ASIDE: The property I used is: (x-y)/(y-x) = -1

So, [2^(-13)][1 - 2^4]/[2^(-17)][2^4 - 1] = (2^4)(-1)

Does that help?

Great strategy!!

We know that, since the numerator is negative and the denominator is positive, the correct answer must be NEGATIVE.
So, the correct answer is either A or B.

Also, the MAGNITUDE of the numerator is greater than the MAGNITUDE of the denominator, which means the value (after the negative sign) must be greater than 1.

Answer: A

Cheers,
Brent

You might want to read my response to Dawnhughan above.

When we perform greatest common factor factoring, we need to find the greatest exponent that each term shares.

For example, let's factor x^7 + x^4 + x^3

In this case, x^3 is the greatest common factor of all three terms.
So, we'll factor it out to get:
x^7 + x^4 + x^3 = x^3(x^4 + x + 1)

Notice that we COULD factor out x^7 as well, but the results are not fun to work with.
x^7 + x^4 + x^3 = x^7[1 + x^(-3) + x^(-4)]

Likewise, if we factor 2^(-13) from the numerator and denominator, we get:
NUMERATOR: 2^(-13)[1 - 2^4]
DENOMINATOR: 2^(-13)[1 - 2^(-4)]

So, our fraction becomes: 2^(-13)[1 - 2^4]/2^(-13)[1 - 2^(-4)]

Simplify to get: [1 - 2^4]/[1 - 2^(-4)]

At this point, our fraction WILL simplify to be -16, but it will take more work to get there than had we factored out the greatest common factor each time.

[1 - 2^4]/[1 - 2^(-4)] = [1 - 16]/[1 - 1/16]
= [-15]/[15/16]
= [-15][16/15]
= -16

Cheers,
Brent

That strategy works, BUT for one very specific reason:

It relies on the following exponent property: [b^(-x)]/[b^(-y)] = (b^y)/(b^x)

So, your approach works (this time!) because we can factor out negative powers in the numerator and denominator and then apply the above property.

Cheers,
Brent

Sorry, I didn't follow that.

In order for the given fraction to equal 1, the numerator and denominator must be equal.

Can you elaborate on your approach?