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.
Last edited by
ceilidh.erickson on Sat Jun 01, 2019 10:28 am, edited 1 time in total.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
