What is the value of m in the equation below?

Don't fall for the trap here! This problem is NOT about the raw math. Many explanations in this forum focus rather blindly on the math, but they fail to realize that the GMAT is a critical-thinking test, not a "let's-see-if-we-can-do-the-math-the-long-way-around" test. Oftentimes, the shape and structure of a problem can point to leverage that allows you to avoid a lot of messy mathematical gymnastics. This problem is no exception.

First of all, let's highlight one of the commonly-found traps embedded in this question. I call it "Mathugliness" in my classes. (Get it? It's math. It's ugly. It acts like a thug. But, like most thugs, it's main game is to intimidate.) The structure of the initial problem is a good example of this principle:

\((\frac{1}{5})^m * (\frac{1}{4})^{24} = (\frac {1}{2(10)^{47}})\)

Now, if you look carefully at this equation, you can see that the left side of the equation tells us that the denominator on the right side is really just a bunch of \(5\)s and \(4\)s multiplied together. And you can't multiply \(4\)s (or \(2\)s) together and have a \(5\) magically pop out. The \(4\)s (or \(2\)s) are completely independent of the \(5\)s here. So don't worry about the \(4\)s or \(2\)s! We are merely looking for the "value of \(m\)" -- which only affects the number of \(5\)s in the denominator. The problem is basically asking, "How many \(5\)s can you factor out of the right side's denominator?" And this isn't hard to figure out.

Since \(10^{47} = (2*5)^{47} = 2^{47}*5^{47}\), there are forty-seven \(5\)s in the right side. We are done. The answer is "D". Doing any more math here is just wasting time.

Now, let’s look back at this problem through the lens of strategy. Your job as you study for the GMAT isn't to memorize the solutions to specific questions; it is to internalize strategic patterns that allow you to solve large numbers of questions. This problem can teach us patterns seen throughout the GMAT. First, this problem is an application of a strategy I call in my classes "Home Bases." The idea is simple: You can’t factor prime bases any further than they already are. If an equation contains identical prime bases on both sides, then the exponents of those bases must also be equal to each other. Thus, if \(x^y = x^z\), then \(y = z\). The next pattern this problem demonstrates is "Mathugliness" -- whereby the GMAT tries to "flex on you," using obnoxious or repetitive math to make the problem seem harder than it actually is. When you encounter Mathugliness, get excited. Look for leverage in the problem that will allow you to solve the problem conceptually, instead of working out all the math. You just need to look at how the problem is structured and determine what would limit or define the value you are looking for. Determine what those limits are, and you have your answer. Patterns turn "inefficient" math into great critical-thinking opportunities. And that is how you think like the GMAT.