The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

5,000 times as large should grow to an ending balance 5,000 times as large. Following

this idea suggests:

End of year 1 balance 
 $5,000(1.08) 
 $5,400.00

which agrees with the result in the table.

What about the next year? Since crediting one year’s interest is equivalent to multiplying

by 1.08, to credit the next year’s interest, we multiply by 1.08 again. Or, in other words, to

get to the second year’s balance we multiply the original balance by 1.08 twice:

End of year 2 balance 
 $5,000(1.08)(1.08) 
 $5,832.00

which once again agrees with the result in the table.

The rationale behind this approach is logical, and the results so far have agreed with

the table. So we have every reason to expect continued success. Using it for 5 years, we

find:

End of year 5 balance 
 $5,000(1.08)(1.08)(1.08)(1.08)(1.08) 
 $7,346.64

which agrees with the table once again. We seem to be on to something here. In fact,

however many years are involved, we can find the amount the account balance will grow

to by repeatedly multiplying by 1.08.

This approach is far more efficient than building an entire table, yet the repeated mul-

tiplications are still tedious. We can accomplish the same thing more efficiently by using

exponents.

An exponent is a way of denoting repeated multiplication of the same number. In

general, xy means y-many x’s multiplied together. For example, 2^3 means three 2s multiplied

together, or (2)(2)(2), which equals 8. Likewise:

(1.08)(1.08)(1.08)(1.08)(1.08) 
 (1.08)^5

Most calculators have a key for exponents; the calculator that you are using with this text

should have one. Different calculator models, though, may mark their exponent keys dif-

ferently. Often this key will be labeled “^”, or “xy”, or “yx”. (Be careful, though; many

calculators also have keys labeled “ex”—this is not the key you are looking for.) Whichever

symbol is used on your calculator, locate this key and let’s try it out. To multiply five 1.08s

together:

1. Enter 1.08


2. Press the exponent key


3. Enter 5


4. Press “
   ” (or “Enter”)

The result should be 1.469328. This was what we multiplied by the original principal, and

so, taking things one step further, if we multiply this result by $5,000, we come up with

$7,346.64, the same as when we went to the trouble of actually doing the multiplication

five times.

Using exponents with a calculator frees us to compound interest over longer periods of

time. For example, we can now find an account balance after 50 years with no more effort

than it took for 5.

(1.08)^50 
 46.901612513

(Your calculator may give more or fewer decimal places, but that is nothing to worry about;

the difference will be insignificant in the final answer.)

Multiplying this result by $5,000, we arrive at $234,508.06, which once again agrees

with the answer we know from the table.

If the interest rate were different, it should be clear that we could use the same logic,

just using the new rate the same way we used the 8%. Likewise, a different principal

3.1 Compound Interest: The Basics 91