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

[I_need_a_700plus]

My apologies if this is a duplicate entry. I had a hard time searching the following GMAT Prep problem within the forum:

If 5^21 * 4^11 = 2 * 10^n what is the value of n?

a) 11
b) 21 (correct answer)
c) 22
d) 23
e) 32

What am I doing wrong?

I changed 4^11 = (2^2)^11 = 2^22

I changed 10^n = (2 * 5)^n = (2^n)(5^n)

New equation: 5^21 * 2^22 = (2)(2^n)(5^n)

Am I approaching this correctly?

[RonPurewal]

That approach is perfect.

Remember that you can consider the power of each different prime separately. (In fact, this is the only reason why prime factorizations are useful.)

So...

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

If you just look at the blue things, you're done. n has to be 21. End of story.

You can also use the not-blue things, but that's a bit more work, since you have to combine 2 * 2^n = 2^(n + 1), and then go with n + 1 = 22.