Life-saving Math Skills

Suppose that . . .

You're stranded on desolate island in the middle of the Pacific Ocean with nothing but an unlabeled canned good and your TI-83 graphing calculator housing dead batteries. A rescue boat filled with Humanitarian Mathematicians finds you and offers to rescue you if you can answer the following question:

Which is bigger, the square root of two or the cube root of three?

What an irrational question involving two "random" irrational numbers! Why on earth would answering a question like that be required for rescue? Remember, these are not just selfless Humanitarians, they are Mathematicians trying to save the world one stranded, computationally-illiterate at a time. Their boat. Their rules.

Do you want to rely on the 50-50 odds of guessing *gulp* incorrectly? Are you brazen enough to ask the Mathematicians if they have four healthy AAA batteries you could borrow temporarily? Is there a way to definitively determine the answer just by doing a couple of quick calculations in the sand? Your life depends on it.

Your instinct might tell you to go with "cube root of three" since, as Shel Silverstein wrote about in "Smart," "three is more than two!" But with any good question posed by a boat load of mathematicians, the answer may be counter-intuitive. In that case, perhaps going with "square root of two. . . final answer?"

The problem is that, although three is bigger than two, taking the cube root of it takes more of the number "away" than taking the square root of it. Therefore, although the two is a smaller number, there is more of it "left" after the indicated operation. Since the numbers two and three are relatively close to each other (with only an infinite number of real numbers between them), the answer is not as obvious as one would think.

Luckily, there is an easy way to answer the question definitively correctly, but it does require a little cleverness, insight, and talent, the same things that enabled Picasso and van Gogh to paint their masterpieces and the same qualities that allowed Bach to compose and Keats to write. The solution is beautifully efficient, artistic, and like a good magic trick, is easy once you know how.

So clear of a space in the sand and grab a shard of seashell and let's work on getting you off the island and into that boat with those creepy Humanitarian Mathematicians.

1. First Let a equal the square root of two and let b equal the cube root of three. Working with single variables will be much easier than working with radical expression .
   
2. Next, rewrite each radical expression as an equivalent expression involving a number raised to a rational exponent.
3. Now comes the clever part. We will change the value of each expression by raising each number to the power of n, where n is the least common multiple of the bases, two and three. In this case, n is 6.
4. Simplifying each new expression using basic rules of exponents (the boatload of mathematicians are very happy that you know when to add exponents and when to multiply them, for sure), we arrive at two easily comparable integers.
5. Now for sure, it is obvious that 9 is greater than 8, so b to the sixth is greater than a to the sixth.
6. Since the numbers are positive, we can now take the sixth root of each side (raising to the 1/6 power) without altering the inequality.
7. Now we arrive at the solution we originally desired. The answer is now so obvious. Since b is greater than a, the cube root of three is in fact bigger than the square root of two.

As you stare at your beautiful artwork, your masterpiece in the sand, you are filled with the satisfaction that you're a genius, a very hungry, dirty, unkempt genius, but a genius nonetheless. As the Humanitarian Mathematicians congratulate you and welcome you on to their boat, you hesitantly leave your creation behind, undisturbed in the sand.

You are saved, hurray. Surviving the boat ride to civilization with a boatload of mathematicians is another story altogether, but at least you saved yourself through your own ingenuity and persistence. It could have been much worse, you know. They could have asked you what was inside the unlabeled can!


By the way, in case your batteries are dead in your calculator too, the decimal values of the two numbers are surprisingly close.