Question: Powers of 4

Comment on Powers of 4

I don't get how when you take common 4^(x-2) then you get 4^2 inside the bracket. plz elaborate.
gmat-admin's picture

Great question! To answer it, consider these more straightforward factoring examples:
k^5 - k^3 = k^3(k^2 - 1)
m^19 - m^15 = m^15(m^4 - 1)
In the first example, we factored out k^3 since 3 is the smaller of the two exponents.
In the second example, we factored out m^15 since 15 is the smaller of the two exponents.
With 4^x and 4^(x-2), the exponents are x and (x-2). Here, the smaller exponent is (x-2), so we'll factor out the 4^(x-2) to get: 4^(x-2) times [something] = 4^x - 4^(x-2)
In this case, we must recognize that [4^(x-2)](4^2) = 4^x, since we must add the exponents, and (x-2) + 2 = x

Does that help?

1Narrative's picture

Hi, Brent! Tried a different approach concerning the numerator : (Any idea why it doesn't work?)

LEFT
4^x - ( 4^x )(4^-2) divided by 5

4^x (1 - 1/4^2) divided by 5

4^x (15/16)(1/5)

RIGHT
= (4^2x)(4^1)
gmat-admin's picture

You're on the right track, but there's a problem with the right side.
It should be 4^(2x+1) - 4^(2x) = (4^2x)[4^1 - 1]

So, we get: (4^x)(15/16)(1/5) = (4^2x)[4^1 - 1]

Simplify: (4^x)(3/16) = (4^2x)[3]

Divide both sides by 3 to get: (4^x)(1/16) = 4^(2x)

Rewrite 1/16 to get: (4^x)[4^(-2)] = 4^(2x)

Simplify left side: 4^(x-2) = 4^(2x)

So, x-2 = 2x

Solve: x = -2

1Narrative's picture

Wow!

Thanks so much. That gave me a sleepless night - lost a bit of confidence right there.

Thanks.

I don't understand how you factorized as the first step? Can you explain as if you were explaining to a layman? (I did see the first comment but still doesnt make sense) :(
gmat-admin's picture

The key is that we always factor out the term with the SMALLEST exponent.

For example, we take x^8 + x^5 + x^11 and factor out x^5.
We get: x^8 + x^5 + x^11 = x^5(x^3 + 1 + x^6)

With 4^x and 4^(x-2), the exponents are x and (x-2).
Here, the smaller exponent is (x-2), so we'll factor out the 4^(x-2)
At this point, the tricky part is determining what goes in the brackets.
When we multiply powers we ADD exponents.
So, [4^(x-2)][4^2] = 4^(x-2 + 2) = 4^x

So, 4^x - 4^(x-2) = [4^(x-2)](4^2 - 1)

Here are three more examples:
5^(x+1) - 5^(x+8) = 5^(x+1)[1 + 5^(x+7)]
2^(x+2) + 2^(x-1) = 2^(x-1)[2^3 + 1]
7^(x-3) + 7^(x-9) = 7^(x-9)[7^6 + 1]

Does that help?

Cheers,
Brent

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