Review of Exponents

1.What are exponents? Exponents are a convenient way of keeping track of the number of factors of one number in a product.
ex. Rather than write
. We agree that a number written like 7^4 (7 to the power 4) means 4 "7"s multiplied together. This is true for a whole number, a fraction or a decimal.


2.PROPERITES OF EXPONENTS:
A. From our definition above we see that

Also
, etc
So we can write

or


Since

Also

Well, if

since we see 2 + 3 = 5 and 1 + 4 = 5, it seems clear that to MULTIPLY numbers together, we must ADD exponents! So, for any number 'x', and whole numbers 'm' and 'n', we have

ex.


Also, suppose we have

We can write this as


But this is the factor "2-squared" times itself 3 times, so


So we can simplify expressions by using the rule:


B. From simple division, we know that 81 divided by 9 equals 9, and 81 divided by 3 is 27. Writing these divisions in fraction form, we see

We conclude to DIVIDE numbers, we SUBTRACT exponents! In general, we have

ex

This last result is a general rule for NEGATIVE EXPONENTS. They equal "1 over". That is,

ex


C. POWERS OF ROOTS

The square root of a number like 9 is another number whose square is 9. That is, it is the number multiplied by itself whose product it 9. Hence, 3 is the square root of 9 because
3 x 3 = 9.
We usually use the "radical sign" to indicate a square root, e.g.,

What we want to do now is to find a power to use instead of the radical sign. This is useful in many types of problems. We want to find the power 'p' so that

Multiplying the 9's on the left of the equal sign (adding exponents), we get

So, 2p = 1 and p must equal 1/2. In other words,


We can find the power for the CUBE ROOT in the same way. That is,

So, we need the power 'p' where

That is, we must have


The fourth root has the power 1/4, etc

D. RATIONAL POWERS

What does the power 3/2 mean? For example, what is the value of

We handle these powers as products. That is, 3/2 = 3 x 1/2. So

Or

Hence

All rational powers can be handled in this way.

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