Understanding Exponents

Date: 10/24/96 at 22:37:55
From: Amanda
Subject: exponents

I just don't get exponents.  My mom and dad have tried to help, but I 
just don't seem to be able to get it.  How do I find the answer to 88 
to the 8th power?  Why would I want to?  It makes no sense!

Date: 01/25/97 at 13:56:41
From: Doctor Donald
Subject: Re: exponents

I'm sorry exponents are confusing you and don't seem to make sense.  
Nevertheless, it is important that you try really hard to understand 
them, because they are an important part of most the math you are 
going to be learning later on.  

How would you calculate 88^8?  Most likely with a calculator.  It is a 
lot of work to do it with pencil and paper, and there is a good chance 
that you would make a mistake because 88^8 is a really big number.  In 
the days before calculators, people used tables of logarithms.

First let's talk about what exponents mean.  88 to the 8th power (we 
write this as 88^8) is the number you get when you multiply 88 times 
itself 8 times.  Written out (where the x's mean times) it is:  

88 x 88 x 88 x 88 x 88 x 88 x 88 x 88   

Really, 88^8 is just shorthand notation for this number since it is 
SO BIG.  Similarly, 2^3 = 2 x 2 x 2 = 8.  What is 2^4?

It is hard to figure out what 88^8 is without a calculater, as I said 
above.  You could make a rough estimate of its value, however, because 
88^8 is less than 100^8, and 100^8 is easy to write down; it's:

 (10^2)^8 = 10^(2*8) = 10^16 = 10000000000000000.  

We see that 88^8 is less than 10000000000000000.  There are 16 0's 
here!  If there were 9 zeros it would be a billion; 12 gives a 
trillion, and 15 a quadrillion.  1 followed by 16 0's is 10 
quadrillion.

Anyhow, 88^8 is about 3596300000000000, according to my calculator, or 
about 3.6 quadrillion.  Note that my calculator won't tell me the 
exact answer to 88^8 since it can't do it exactly -- it doesn't have 
enough memory.  If you did the multiplication yourself with pencil and 
paper, you wouldn't have enough memory either but you would use the 
paper to act as your memory, and you can use any amount of paper to 
keep track of the calculation.  The calculator has very little 
"electronic paper", only enough to remember 7 or 8 digits at a time.

So you, as a human being, could actually (in principle) get the exact 
answer to 88^8, beating the calculator.  I don't think it would be 
much fun, however.  Other calculators with more memory (and computers, 
etc) can also get the exact answer.

So why would you care about 88^8?  Well, that's hard to say.  Let's 
see... a piano has 88 keys.  If you just went up to the piano and hit 
one key after another 8 times (just choose the keys at random - close 
your eyes, spin around, hit a key, do it again,...), how many 
different "songs" could you play?  The answer is EXACTLY 88^8.  That's 
a lot.  Of course, they wouldn't sound like music, though given the 
variety of things people call music, maybe that's not true! 

To make the piano experiement more reasonable, you could just restrict 
yourself to hitting the keys in one octave - the 88 keys on the piano 
are all just A,A#,B,C,C#,D,D#,E,F,F#,G,G#, so you could take your 
random song and transpose it to a single octave.  It might sound 
better.  How many such songs are there?

It turns out exactly 12^8, or about 429980000. There are about 429 
million different 8-note melodies which stick to a single octave, 
starting at A, say.

Calculations like this explain why there seems to be no limit to the 
number of musical compositions, and also why if you hit keys at random 
it isn't likely to sound familiar.  

I don't know whether this has interested you at all, but I think that 
your problem with exponents is not so much that you are intellectually 
incapable of learning about them, but rather that the job seems 
pointless.  Math is a lot easier to do when it is fun, even if it in
fact quite "difficult".  

I hope this helps a little.

-Doctors Donald and Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/