# Some Exponent Practice

My son had two problems from his Mathcounts practice that puzzled him last night. I thought it was best that he puzzled through one of the questions – what is 47 * 53. His question about this problem was simple – why would they even ask this question?

The other question required a little more review since we’ve not talked about exponents in a while:

Evaluate: $( ( \sqrt{2} )^{\sqrt{2}})^{\sqrt{8}}$

Unlike the first problem which really just involves learning a difference of squares trick, this problem actually connects to an interesting puzzle (and one that we’ve studied before):

Can an irrational number to an irrational power be rational?

I didn’t go all the way to exploring that connection last night, but instead we had a nice 10 minute review about exponents.

By luck, today he was working through a few problems from the 2010 AMC 8 and came across this problem:

Problem #24 from the 20010 AMC 8

The problem asks you to arrange these three numbers in increasing order – $10^8 , 5^{12}$, and $2^{24}$. A nice problem to encounter after a review of exponents.

Here’s how he approached it:

After he worked through the problem I wanted to do a quick review just to give a little extra emphasis to last night’s review.

So, a happy coincidence running into this problem today. Nice to be able to put our little exponent review from last night to work today!