need help on a question

[aiq]

hello everybody.....
i am from ohio and i just found this awesome forum..... i took gmat in january last year with one week preparation and got 570. i am studying using manhattan and OG and the last prep test i got 640.... so i m hopeful i will do better this time round....
now i have a question:

5^21 * 4^11 = 2 * 10^n, what is the value of n?
a)11
b)21
c)22
d)23
e)32

thanks a ton in advance....

[piyushdabomb]

The answer should be 21.

Here it goes:

1) 5^21 * 4^11 = 2 * 10^n (Question)
2) (5^21 * 4^11)/2 = 10^n
3) (5^21 * 4 * 4^10)/2 = 10^n
4) (5^21 * 2 * 4^10) = 10^n
5) (5^21 * 2 * 2^20) = 10^n
6) (5^21 * 2^21) = 10^n
7) 10^21 = 10^n
8) Therefore, n = 21

Use these rules when in doubt:

https://www.mathwords.com/e/exponent_rules.htm

[Ian Stewart]

hello everybody.....

5^21 * 4^11 = 2 * 10^n, what is the value of n?
a)11
b)21
c)22
d)23
e)32

thanks a ton in advance....

A good solution above. There are a few ways to get to an answer here, and when you see these kinds of equations, with numbers and large exponents, one foolproof method is to break everything down into primes:

5^21 * 4^11 = 2 * 10^n
5^21 * (2^2)^11 = 2 * (2*5)^n
5^21 * 2^22 = 2 * 2^n * 5^n
5^21 * 2^22 = 2^(n+1) * 5^n

and since we have prime factorizations on both sides, the exponents must be equal on both sides - n = 21.

It's a good question, and illustrates how important it is to have a good command of exponent rules.