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Exponent Time 2 (GPREP)

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by zagcollins » Wed Jul 30, 2008 9:16 pm
5^21*4^11=2*10^n Find value of n

Cant remember the answer options as usual..

this is how i solved it...

5^21*2^11*2^11=2*10^n
Getting 2 in RHS to LHS changes the equation to 5^21*2^10*2^11=10^n

which is nothing but 5^21*2^21=10^n
which can then be rewritten as 10^21=10^n
Cancelling the 10's, n=21.

IS the answer correct and is there a faster method??
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Re: Exponent Time 2 (GPREP)

by Stuart@KaplanGMAT » Wed Jul 30, 2008 9:30 pm
zagcollins wrote:5^21*4^11=2*10^n Find value of n

Cant remember the answer options as usual..
Let's make the left side look like the right side.

5^21*4^11
5^11 * 5^10 * (2*2)^11
5^11 * 5^10 * 2^11 * 2^11
(5^11 * 2^11) * (5^10 * 2^10) * 2^1
(5*2)^11 * (5*2)^10 * 2
10^11 * 10^10 * 2
10^(10+11) * 2
10^21 * 2 = 2*10^21
n = 21

We also could have started by dividing both sides by 2, to get:

(5^21*4^11)/2=10^n

and solved from there - this way the extra "2" would have cancelled out earlier, so we'd eventually get:

(5^11 * 2^11) * (5^10 * 2^10) = 10^n

and proceed from there to end up with:

10^21 = 10^n
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Re: Exponent Time 2 (GPREP)

by Ian Stewart » Thu Jul 31, 2008 2:51 am
zagcollins wrote:5^21*4^11=2*10^n Find value of n

Cant remember the answer options as usual..

this is how i solved it...

5^21*2^11*2^11=2*10^n
Getting 2 in RHS to LHS changes the equation to 5^21*2^10*2^11=10^n

which is nothing but 5^21*2^21=10^n
which can then be rewritten as 10^21=10^n
Cancelling the 10's, n=21.

IS the answer correct and is there a faster method??
Yes, that is correct, and is a good solution. There is a faster method which is a bit more abstract, but I don't see anything wrong with your approach, as long as you can do it within in two minutes. Anyway, you can also look at it as follows:

We're dealing with integers, so in the prime factorization of the left and the right sides of the equation, the primes and their exponents must be equal. You have 5^21 on the left, and you would get 5^n on the right if you divided up the 10, so n must be 21.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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