Large exponents

BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

This topic has expert replies

Post new topic Post Reply

Previous Topic Next Topic

cooperbaker: Junior | Next Rank: 30 Posts; Posts: 10; Joined: Mon Dec 15, 2008 6:27 pm; Location: Atlanta, GA

Large exponents

by cooperbaker » Sun Jan 25, 2009 1:48 pm

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

OA: D

I didn't know how to solve this question using the properties of exponents. What I did realize is I could subtract 10 from all the exponents then use real numbers in my head -- (i.e. 3(2^3) = 24)

Is this a good way to go about it, or is this wasting time? My $40 Kaplan book doesn't explain what to do when adding or subtracting large exponents with the same base....

Quote

Source: Beat The GMAT — Problem Solving | Sun Jan 25, 2009 1:48 pm

truplayer256: Master | Next Rank: 500 Posts; Posts: 392; Joined: Thu Jan 15, 2009 12:52 pm; Location: New Jersey; Thanked: 76 times

by truplayer256 » Sun Jan 25, 2009 1:57 pm

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

2^x-(2^x/2^2)=3(2^13)
2^(2+x)-2^x=(2^15)(3)
2^x(2^2-1)=(2^15)(3)
2^x(3)=2^(15)(3)
3's cancel
2^x=2^15
x=15 D

Quote

cooperbaker: Junior | Next Rank: 30 Posts; Posts: 10; Joined: Mon Dec 15, 2008 6:27 pm; Location: Atlanta, GA

by cooperbaker » Sun Jan 25, 2009 2:02 pm

truplayer256 wrote:
If 2^x - 2^(x-2) = 3(2^13), what is the value of x?
2^x-(2^x/2^2)=3(2^13)
2^(2+x)-2^x=(2^15)(3)
2^x(2^2-1)=(2^15)(3)
2^x(3)=2^(15)(3)
3's cancel
2^x=2^15
x=15 D

You lost me at step 1...

2^(x-2) = 2^x/2^2 ???

Quote

truplayer256: Master | Next Rank: 500 Posts; Posts: 392; Joined: Thu Jan 15, 2009 12:52 pm; Location: New Jersey; Thanked: 76 times

by truplayer256 » Sun Jan 25, 2009 2:07 pm

by rules of exponents, if the numbers have the same base,and you're diving them, you can subtract the exponents..so, 2^x/2^2=2^x-2, since 2^2 and 2^x have similar bases.

Quote

Zipper: Master | Next Rank: 500 Posts; Posts: 119; Joined: Sun Nov 30, 2008 2:47 am; Thanked: 11 times

by Zipper » Sun Jan 25, 2009 2:52 pm

Here are a couple of simpler solutions for this problem.

https://www.beatthegmat.com/3-problems-f ... 27186.html

Quote

GMAT/MBA Expert

Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 am; Location: Montreal; Thanked: 1090 times; Followed by:355 members; GMAT Score:780

Re: Large exponents

by Ian Stewart » Sun Jan 25, 2009 2:54 pm

cooperbaker wrote: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

OA: D

I didn't know how to solve this question using the properties of exponents. What I did realize is I could subtract 10 from all the exponents then use real numbers in my head -- (i.e. 3(2^3) = 24)

Is this a good way to go about it, or is this wasting time? My $40 Kaplan book doesn't explain what to do when adding or subtracting large exponents with the same base.... :?

Your method here is clever, just as long as you add 10 back at the end. I'd be a bit concerned that you'd have trouble doing the same with similar problems, however. There are a few ways to solve the equation as provided; truplayer gives one good approach above, and I'll suggest another. In general, when you're adding or subtracting terms with the same base and different powers, you'll want to factor out the term with the smallest power. For example, if you see x^5 - x^3, in most cases it will be useful to factor out x^3, to get:

x^5 - x^3 = x^3(x^2 - 1)

(and depending on the question, you might want to factor the x^2 - 1 = (x+1)(x-1) as well). In the question in the original post, the smaller power is (x-2), so we can factor out 2^(x-2), to get:

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

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com

Quote

awilhelm: Master | Next Rank: 500 Posts; Posts: 117; Joined: Fri Oct 17, 2008 7:41 am

by awilhelm » Sun Jan 25, 2009 7:59 pm

Could anyone please explain why my approach doesn't lead to the correct answer?

2^x - 2^(x-2) = 3(2^13)
2^x - 2^x * 2^-2 = 3(2^13)
2^x(1 - 2^-2) = 3(2^13)
2^x(1 - 1/4) = 3(2^13)
2^x(3/4) = 3(2^13)
4 * 2^x * 3 = 3(2^13) * 4
12 * 2^x = 12 * 2^13
2^x = 2^13
x = 13

Quote

GMAT/MBA Expert

Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 am; Location: Montreal; Thanked: 1090 times; Followed by:355 members; GMAT Score:780

by Ian Stewart » Sun Jan 25, 2009 8:56 pm

awilhelm wrote:Could anyone please explain why my approach doesn't lead to the correct answer?

2^x - 2^(x-2) = 3(2^13)
2^x - 2^x * 2^-2 = 3(2^13)
2^x(1 - 2^-2) = 3(2^13)
2^x(1 - 1/4) = 3(2^13)
2^x(3/4) = 3(2^13)
4 * 2^x * 3 = 3(2^13) * 4
12 * 2^x = 12 * 2^13
2^x = 2^13
x = 13

It's almost entirely right, and it will lead to the right answer if you fix one small error. In the line highlighted in red above, you intended to multiply both sides by 4, but on the left side, you've actually multiplied by 16 -- you cancelled the 4 in '3/4' but you've introduced an additional 4 as well.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com