GMAT Math Tip: The powers that be

There are a few tricky ways that the GMAT tests your exponent acrobatics. Specifically, the concept of “regrouping powers” is fairly common.

The typical question goes something like this:

3^3 * 4^3 * 5^3 = 60^n
n = ?

Here’s how a solution looks:

3^3 * 4^3 * 5^3 = 3*3*3 * 4*4*4 * 5*5*5 = (3*4*5) * (3*4*5) * (3*4*5) = (3*4*5)^3 = 60^3
n = 3.

Here’s another example (got this in an email from one of my students today):

5^21 * 4^11 = 2 * 10^n
n= ?

“What’s the intuition here?” –A

Here’s how to approach this type of problem:

The intuition here is to make left and right side look similar. Specifically, you need to come up with a 10^x on the left side. Thinking of the left side as a string of 21 5′s followed by a sting of 11 4′s, one way to go is to say that 11 4′s is like 22 2′s (ie, (2×2)(2×2)… 22 times). Now we can alternate the 5′s and 2′s 21 times (ie (5×2)(5×2)… etc) and have one 2 left over. Doing so, you get 10^21×2 on the left side. Therefore n has to be 21.

Answer is: 2 * 10^21