Exponent Question

Whenever you're given variable exponents, your goal will be to set the bases equal, so you can set the exponents equal.

For example, if \(3^x = 81\)
then \(3^x = 3^4\)
therefore x = 4

In most problems, you want to simplify to prime bases.

In the problem given, reduce everything to primes:
$$5^{21}\cdot4^{11}=2\cdot10^n$$
$$5^{21}\cdot\left(2^2\right)^{11}=2\cdot10^n$$
$$5^{21}\cdot\left(2^2\right)^{11}=2\cdot\left(2\cdot5\right)^n$$
$$5^{21}\cdot2^{22}=2\cdot\left(2^n\cdot5^n\right)$$
$$5^{21}\cdot2^{22}=2^{n+1}\cdot5^n$$

Since our bases are equal on both sides, we can set our exponents equal:
If \(5^n = 5^{21}\), then n = 21
If \(2^{n+1} = 2^{22}\), then n = 21

The answer is B.

For more on solving for variable exponents, see:
https://www.beatthegmat.com/2-x-2-x-2-3- ... tml#578272
https://www.beatthegmat.com/manhattan-qu ... tml#662728