Algebra Know-It-ALL

Part One 163

To make the original number an order of magnitude smaller, we divide by 10. That can be
done by moving the decimal point to the left by one place and then repositioning the comma.
We end up with

3,546.80337

Question 7-10

A negative integer power is a nonzero quantity divided by itself a certain number of times.
How can we show the meaning of this by examples in the decimal system?

Answer 7-10

Let’s start with the number 10. This is 10^1. When we divide 10 by itself, we get 10^0 , which is 1.
When we divide 10 by itself twice, we get 10−^1 , which is 0.1. It goes on like this for negative
integer powers:

10 divided by itself three times is 10−^2 , which is 1/10^2 or 0.01

10 divided by itself four times is 10−^3 , which is 1/10^3 or 0.001

10 divided by itself five times is 10−^4 , which is 1/10^4 or 0.0001

↓

and so on, as far as we want!

Chapter 8

Question 8-1

Suppose we start with 7 and raise it to the −1st power, then the −2nd power, then the
−3rd power, then the −4th power, then the −5th power, and so on, endlessly. What happens
to the result?

Answer 8-1

We get a sequence of fractions that converges toward 0. It starts at 1/7, and then keeps getting
1/7 as large with each succeeding power. Here’s what happens:

7 −^1 = 1/7

7 −^2 = 1/7^2 = 1/49

7 −^3 = 1/7^3 = 1/343

7 −^4 = 1/7^4 = 1/2,401

7 −^5 = 1/7^5 = 1/16,807

↓

and so on, forever