Natural Blogarithms

Why didn’t I think of that?

November 24th, 2008 by SplineGuy

Problem: Which is larger, $2.2^{3.3}$ or $3.3^{2.2}$ ?

Sure, there’s the easy way: stick it in your calculator. So, make it interesting. Answer the question without using the calculator.

This raises a couple of interesting questions:

1. What do we mean by raising a number to a (terminating) decimal or fractional power?

2. Can I do that without a calculator?

The answer to the first question is, of course, roots. The exponents above can be written as fractions, and we learn in algebra that

$\displaystyle a^{\frac{m}{n}} = \sqrt[m]{a^n}$

So, we interpret $2.2^{3.3}$ as

$2.2^{\displaystyle \frac{33}{10}} = \sqrt[10]{2.2^{33}}$

Similary, we interpret $3.3^{2.2}$ as

$3.3^{\displaystyle \frac{11}{5}} = \sqrt[5]{3.3^{11}}$

What about question 2, then? The answer is, I certainly can’t. There is a fairly easy-to-use algorithm (step-by-step) procedure for calculating square roots by hand. A simple Google search reveals many places to find the algorithm explained. However, the only method I would try for calculating a 10th root or, even, a 5th root by hand might be a series approximation of each root function. You will quickly decide that…

There must be an easier way:

This problem was originally posted over at the Wild About Math! blog that I read regularly. I worked on a few things, off and on for a few days and failed to come up with anything that didn’t require me to multiply 2.2 or 3.3 out 33 or 22 times, respectively. Fortunately, there was a solution posted today and made slap my forehead and say, “Man, I wish I had thought of that…”

(By the way, slapping the forehead should be avoided during a migraine cluster, such as I have been dealing with the last four days)

Here’s the posted solution:

$2.2^{3.3} = (2.2^3)^{1.1}$

$\ \ \ = (2.2 \times 2.2 \times 2.2)^{1.1}$

$\ \ \ = (4.84 \times 2.2)^{1.1}$

$\ \ \ = 10.648^{1.1}$

We can also say the following,

$3.3^{2.2} = (3.3^2)^{1.1}$

$\ \ \ = (3.3 \times 3.3)^{1.1}$

$\ \ \ = 10.89^{1.1}$

Because the function $f(x)=x^{1.1}$ is a monotonically increasing function for $x>0$ , then we can say $3.3^{2.2}$ is larger since $10.89^{1.1}>10.648^{1.1}$