Algebra 2: Exponents

Date: 01/07/98 at 01:38:49
From: Maryann
Subject: Algebra 2 - Exponents

I need to find out what "x" is in this problem: x^x^3 = 3. I figured 
out the answer, but by using solver on my calculator. Now the teacher 
told us to find out how to solve the equation without using solver or 
graphing. I have no idea how to do that and what to do to begin.

Date: 01/07/98 at 08:34:32
From: Doctor Mitteldorf
Subject: Re: Algebra 2 - Exponents

Dear Maryann,

In this problem, you happen to be "lucky." If you happen to notice 
that x^3 = 3 makes the exponent 3 also, so the cube root of 3 solves 
the equation. 

You could leave it at that. But it's interesting to think... what if 
the 3 in the exponent and the 3 on the right had been two different 
numbers - say 3 and 2. What if your equation was x^(x^3)=2 - how would 
your solve that

There's no integer like 2 or 7 that solves this equation. There's no 
rational number like 4/3 or 167/125 that solves it either. And I don't 
think that you can write the solution as the square root or cube root 
of anything. So it's just a continuing decimal 1.336..., and the best 
you can do is to find a calculation method that keeps getting you more 
and more digits of the answer.

So... what's a good calculation method? A standard one is called 
Newton's Method; it works in lots of problems like this one, but it 
requires calculus, and I think I can offer you one that is less 
general, but doesn't need calculus.  It's called "iteration."

You can write the equation as x^(x^3) = 2. Taking the x^3 root of both 
sides, you can write it as x = 2^(1/x^3). You can use this form of the 
equation to get better and better guesses for x. Say your first guess 
is 1.5. Use this number in the right side of the equation, and take 
2 to the power 1/(1.5)^3. The answer is x=1.2279...  This number is 
closer to the answer than the 1.5 that you started with. Now do it 
again: take 2 to the power 1/(1.2279)^3.  The answer now is 1.454.

Each time you do this, you get a little closer to the answer. You 
could do it on a calculator or a computer, and after 20 or 30 steps, 
you'd be as close as you want to be.

Even better: notice that the result keeps jumping up and down, 
estimating too high, then too low, then too high again. When this 
happens, you can usually do better by averaging each result with the 
last. For example, if your first x was 1.5 and the next was 1.2279, 
you'd average those two to give 1.364 as your next guess. Then take 2 
to the power 1/(1.364)^3, the answer is 1.307, average this with 1.364 
and get 1.335 which is already very close to the right answer.  
Averaging helps you to get closer with a lot fewer repetitions.

Try this method with x^(x^3)=3 instead of =2, and see if you can get 
closer and closer to the cube root of 3 as an answer.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/