Problem Solving

G- Unit30 wrote:Hi,
I had this problem on GMATprep practice exam, so I don't have the full list of anwsers but the question is:

What is the value of x?
2^x - 2^(x-2) = 3(2^13)

neeg wrote:I'm unclear about the very fist step of factoring
2^(x-2)[2^2 - 1] = 3(2^13)....how is 2^(x-2)a common factor?

Good question.
To answer this, let's examine some analogous factoring cases:

k^5 - k^3 = k^3(k^2 - 1)
m^19 - m^15 = m^15(m^4 - 1)

Or how about:
w^x + x^(x+5) = w^x(1 + w^5)

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

So, in the expression 2^x - 2^(x-2), the term with the smallest exponent is 2^(x-2), so we'll factor out 2^(x-2)

Does that help?

Cheers,
Brent

neeg wrote: Thanks Brent.

I understand for the below two as they have common base raised to the same powers and hence can be taken out as the common factor.
k^5 - k^3 = k^3(k^2 - 1)
m^19 - m^15 = m^15(m^4 - 1)

However, for w^x + x^(x+5) = w^x(1 + w^5)-----w^x does not appear in the second term at all so how does it become a common factor?

Similarly for the expression 2^x - 2^(x-2), I understand that smallest exponent is 2^(x-2)but don't see how this is a common factor in the first term which is 2^x?

J N wrote:
does this make sense:

2^10 - 2^6 = whatever

pull out 2^10

2^10 (1 - 1^-4) ????

2^10 (1-1) ??????

=0 ?????