5^21*4^11=2*10^n

[Vali]

5^21*4^11=2*10^n

a. 11
b. 21
c. 22
d. 23
e. 32

Please explain every step.


Source: Gmat Prep.

[Guest]

Hi Vali -

5^21*4^11=2*10^n

First step - recognize that 4^11 = (2^2)^11 = 2^22 so:

5^21 * 2^22 =2*10^n

Second Step: note that 10 is equal to 5*2, so the left side of the equation gives you 21 factors of 10, with one 2 left over:

2*10^21 = 2*10^n, so you can see that n=21
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The solution above featured a shortcut, of actually combining prime exponents. A method that will always work is to reduce everything to prime exponents instead.

5^21 * 4^11 = 2*10^n

5^21 *(2^2)^11 = 2*5^n*2^n : Recognize that 4 = 2^2 and 10 = 2*5

5^21 * 2^22 = 2^(n+1) * 5^n

At this point, you can either the exponents of the 5 to see n=21, or the exponent of the 2's to see n+1=22, n=21.

Cheers,

Jeff

[StaceyKoprince]

Nice explanation Jeff.

[vali]

Thank you,