1. ## GMAT question

Hi everyone,

So for several months I have been busy studying for the GMAT which I'll be taking this coming Saturday. Today I took a prep. test and I am just blown away by how hard it is. The prep test is much harder than the math problems that I solved in the books I have been studying. I was wondering if you could help me solve a couple (or more) of these problems that are giving me a really hard time so that I could be as prepared as I possibly can. Thanks in advance for any help.

Q: If 5^21 x 4^11 = 2 x 10^n, then what is n?

Just in case...the ^ symbol represents exponent. Using a calculator I can work this out, but in the test I can't use a calculator. Anyone know of a shortcut? Thanks.

2. Originally Posted by crazirani

Q: If 5^21 x 4^11 = 2 x 10^n, then what is n?

Just in case...the ^ symbol represents exponent. Using a calculator I can work this out, but in the test I can't use a calculator. Anyone know of a shortcut? Thanks.
Prime factorize,
$\displaystyle 4^{11}=(2^2)^{11}=2^{22}$
And,
$\displaystyle 10^n=(2\cdot 5)^n=2^n\cdot 5^n$.
Thus,
$\displaystyle 5^{21}\times 4^{11}=5^{21}\times 2^{22}$
And,
$\displaystyle 2\times 10^n=2\times (2\cdot 5)^n=2\times 2^n \times 5^n=2^{n+1}\cdot 5^n$
Thus,
$\displaystyle 5^{21}\times 2^{22}=5^n \cdot 2^{n+1}$
By just looking at the exponents we see that,
$\displaystyle n=21$

3. That's amazing! Thanks a lot.

EDIT: I just realized that I posted this in the "Elementary and Middleschool" folder...sorry.

4. Originally Posted by crazirani
Hi everyone,

So for several months I have been busy studying for the GMAT which I'll be taking this coming Saturday. Today I took a prep. test and I am just blown away by how hard it is. The prep test is much harder than the math problems that I solved in the books I have been studying. I was wondering if you could help me solve a couple (or more) of these problems that are giving me a really hard time so that I could be as prepared as I possibly can. Thanks in advance for any help.

Q: If 5^21 x 4^11 = 2 x 10^n, then what is n?

Just in case...the ^ symbol represents exponent. Using a calculator I can work this out, but in the test I can't use a calculator. Anyone know of a shortcut? Thanks.
Here is one way.

(5^21)(4^11) = 2(10^n)
There's a 2 at the RHS and a 4 at the LHS, so express the 4 in terms of 2,
(5^21)(2^22) = 2(10^n)
Divide both sides by 2,
(5^21)(2^21) = 10^n
(5*2)^21 = 10^n
10^21 = 10^n