For exponent questions, look for ways to work with the same base or bases, if you can. Also look for clues in the values given in the problem. Because we're given numbers like 2, 81, and 18, there's a good chance that the best strategy will have something to do with 2's and 9's (81=$9^2$ and 18 = 2*9)

Now we'll use some exponent rules to rewrite the equation in a way that's easier to work with. Since there's already a large value in a denominator that we're clearly not meant to calculate ($18^24$, we'll make life easier by rewriting each expression so that the power is in the denominator across the board:

$(\frac{1}{2})^{24}$=$\frac{1^{24}}{2^{24}}=\frac{1}{2^{24}}$ and $(\frac{1}{81})^k=\frac{1^k}{81^k}=\frac{1}{81^k}$

So

$\frac{1}{2^{24}}*\frac{1}{81^k}=\frac{1}{18^{24}}$

Now, rewrite 81 as $9^2$ so that we have similar bases we can work with:

$\frac{1}{2^{24}}*\frac{1}{(9^2)^k}=\frac{1}{18^{24}}$

Since the numerator is 1 all the way across, we can disregard it: $2^{24}*9^{2k}=18^{24}$

Divide both sides by $2^{24}$ and we're left with $9^{2k}=(18/2)^{24} =9^{24}$

Since the base (9) is the same, we can disregard it:

$2k=24$ or $k=12$

Answer: B