[Math] exponent manipulation – $4^{21} \cdot 5^{11} = 2 \cdot 10^{n}$ – what is $n$

exponentiation

I was taking a practice GMAT test and it had a question like this:

$4^{21} \cdot 5^{11} = 2 \cdot 10^n$

What is $n$?

The available answers were something like
16
22
23
24
32

I'm not exactly sure on the multiple choice options…

However I'm not allowed a calculator so I'm not sure how to go about this…

I'm generally pretty good at the little tricks that allow you to solve a questions without the calculator but I have no idea how to do this one…

Thanks!

Update:

I found the actual question:

enter image description here

Best Answer

I'm assuming you meant $4^{11}\cdot 5^{21} = 2\cdot 10^n$. The rules you want to know for exponents can be found on http://en.wikipedia.org/wiki/Exponentiation (this page has a lot of information, so depending on what you need, you may only be interested in http://en.wikipedia.org/wiki/Exponentiation#Identities_and_properties ). For this question, in particular, you want \begin{align*} (a\cdot b)^n &= a^n\cdot b^n\ \ \operatorname{and}\newline a^{nm} &= (a^n)^m \end{align*}

So looking at $10^n = (2\cdot 5)^n$, we get \begin{align*} 4^{11}\cdot 5^{21} &= (2^2)^{11}\cdot 5^{21}\newline &= 2^{22}\cdot 5^{21}\newline &= 2\cdot (2^{21}\cdot 5^{21}). \end{align*}

So if you question was with the exponents reversed on the 4 and 5, we get that $n=21$.

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