Exponents and Averages (two problems help)

Question

Hey folks,

I answered these two question on the GMAT Prep official practise exam #1, I got both answers wrong, but I don't even understand why the right answers are so

First one is :

2^(4-1)^2
-----------
2^3-2

Their answer is 2^8, why ?

The second one is a simple average problem, and I can't believe I got it wrong. It goes something like :

Susan drove the first 30 miles and an average speed of 30 mph, and the remaining 30 miles at an average speed of 60 mph, if Susan didn't make any stops, what was her average speed for the entire trip ?

Their answer is 40, I can't see why

I realize with average sometimes the weights are different (like over a year he average of games played the first 4 months was 30 per month, and over the other 8 months it was 20 per months), this one is not a simple add up and divide per two, obviously weight should be taken into account. But in the Susan driving example, both were covering an equal distance of 30 miles, in an unequal time obviously

Thanks for your help

nsp007 · Answer

1. I assume it is : \frac{2^{{(4-1)}^2}}{2^{(3-2)}}

--> \frac{2^{3^2}}{2}  (it is 3^2)

-->  \frac{2^9}{2}

-->  2^8

nsp007 · Answer

2. Avg. speed = total distance / total time

first 30 miles at an average speed of 30 mph --> time = 1hr

remaining 30 miles at an average speed of 60 mph --> time = 1/2 hr

Avg. speed =  \frac{(30 + 30)}{(1+0.5)}  = 40mph !

kuharido · Answer

Thanks, though i still didnt really understand

the problem on the exam is written as you typed, with the stuff between parenthesis in superscript, but the numbers in the denominator don't have any paranthesis, here is a better detail

Numerator : 2 superscript (4-1)^2

Denominator : 2 superscript 3-2

I am assuming the first superscript is a power, and ^ is a power of that, in that case, shouldn't the numerator be equal to

2^ 3 ^ 2 = 2^6 ?

and the denominator = 2^3-2 = 2^1 = 2 ?

2^6/2 = 2^5 shouldnt this be the answer ?

kuharido · Answer

Ok i got the average one, dont know why i averaged both values !

Its the first prep test i took and i got a 40q score without any studying or practise, i'm pretty bummed as I don't know by how many points would practise improve it anyways, it seemed like a you know it or you dont kind of thing

kuharido · Answer

Oh sorry ok, i got the exponents one as well now, jeez whats wrong with me

Thanks man i really appreciate it

So when its 2^3^2 i thought you multiply the exponents (3*2=6) and then it becomes 2^6 no ? am i wrong ? or is this case an exceptional notation ?

nsp007 · Answer

I am assuming the first superscript is a power, and ^ is a power of that, in that case, shouldn't the numerator be equal to

2^ 3 ^ 2 = 2^6 ?

2^3^2 is not the same as (2^3)^2.

2^3^2 = 2^9
(2^3)^2 = 8^2 = 2^6

Hope this helps !

kuharido · Answer

It does, mega thanks friend

Bunuel · Answer

Solution provided by nsp007 is correct:

If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:
 a^m^n=a^{(m^n)}  and not  (a^m)^n , ( (a^m)^n=a^{mn} ).

So  2^{{(4-1)}^2}=2^{3^2}=2^9 .

Check Number Theory chapter of the Math Book for more on this issue (link in my signature).

Hope it helps.

kuharido · Answer

Bunuel thanks a lot, especially for the guide ! Actually I had already bookmarked that treasure of math knowledge but didn't get to review it, thank you P.S "exponentiation" is fun to say

kuharido · Answer

I realize this isn't the best post to ask this on but, do you reckon an 8 point increase is within reach with proper practice (from 40 to 48) ? (I have about two months, can put in good hours every week)

Exponents and Averages (two problems help)

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