Exponents

[Holden123]

If 2^x - 2^(x-2) = 3(2^13), what is the value of x ?

Can someone please explain!

[Brent@GMATPrepNow]

If 2^x - 2^(x-2) = 3(2^13), what is the value of x ?

This requires some factoring. We'll factor 2^(x-2) from the left side.
2^x - 2^(x-2) = 3(2^13)
2^(x-2)[2^2 - 1] = 3(2^13)
2^(x-2)[3] = 3(2^13)
So, 2^(x-2)= 2^13
x-2 = 13
[spoiler]x = 15[/spoiler]


Aside: Some students don't see how we can factor 2^(x-2) from the left side. So, let's first examine some analogous factoring cases:

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


Cheers,
Brent

[Holden123]

Thank you!!!

[GMATGuruNY]

When you post a problem, please include the answer choices.

If 2^x - 2^(x-2) = 3(2^13), what is the value of x?
a) 9
b) 11
c) 13
d) 15
e) 17

We can plug in the answer choices for x.

Answer choice C: x= 13
2^13 - 2^(13-2) = 3(2^13)
2^13 - 2^11 = 3(2^13)
2^11(2^2 - 1) = 3(2^13)
2^11(3) = 3(2^13)

Plugging in x=13 made the exponent on the left 2^11.
To match 2^13 on the right side of the equation, the exponent needs to be increased by 2.
Thus, x = 13+2 = 15.

The correct answer is D.

Algebraically:
Try get SIMILAR BASES on each side of the equation.
Given x^y ± x^z, factor out the SMALLER EXPONENT.

2^x - 2^(x-2) = 3(2^13)

2^(x-2)(2^2 - 1) = 3(2^13)

2^(x-2)(3) = 3(2^13)

2^(x-2) = 2^13

x-2 = 13

x = 15.