Properties of Exponents

The following properties apply to exponents:

1. (bases are the same and we are multiplying the two together , then we
add the exponents ).
2. (When we are raising it to a power, we multiply the exponents).
3. (This is like the distributive property of exponents, even though that
name doesn't exist, we still treat it act like. We apply the exponents to everything inside
of the parenthesis .
4. (again similar to property 3, we apply the exponent to everything).
5. ( anything to the zero power is always equal to one)
6. (notice again the bases are the same on the top and bottom)
7. (anytime we have a fraction to a negative number, we flip the entire
fraction and the negative exponent becomes positive. I actually put a negative one in this
problem but it could be any negative number).
8. (whenever we have anything to a negative power, we can take it
opposite of where it is….If it is in the denominator, we take it to the numerator, if it is in
the numerator, we take it to the denominator…..once we take it opposite of where it is, it
becomes a positive exponent ). This doesn't apply to negative numbers, just negative
9. (notice here we didn't have a fraction to begin with, so our first step
is to write it in fraction form and then take it opposite of where it is. Notice on property 8
and 9, I have actually plugged in numbers, these can actually be any negative number!

Before we dive in, lets review radicals a little bit. The number in the little slot
in the front of the radical is called our index.

, our index is 3
, our index is 4
, our index is 2 (note if there is no number, the index will be 2 by

Our index tells us how many times a number ( or variable ) should be
multiplied by itself before we can take it out of the radical.


Since our index was 2 and we found that 9 times 9 gives us 81, we brought the
9 out.


Since our index was 3 and we found 2 times itself three times gives us 8, we
could bring a 2 out.

Sometimes, we have to leave things in the radical:


Notice in this example, we didn't have a pair of threes to bring out!
Again, you have seen this material before, the difference is that now we are going to mix
in radicals before we do these operations . The first thing we need to look at is how to go
from radical form to exponential form:




(remember if there is no number for the index , it is 2 by default!)


(remember if there is no power on something, it is the invisible 1 by default)


(just like you would expect, in reverse)

Radicals are bad enough but fractional exponents, that’s even worse! We do it so we can
use our properties above.


Even though fractions aren't that nice, they are infinitely better than working with
radicals because we can take advantage of all our properties of exponents.

Simplifying a Number that has a fractional exponential with a number greater than one
in the numerator


We always split these up. We put the numerator on the outside of a pair of parenthesis:

This is going backwards from the property that allows us to multiply exponents together
when raising it to a power. We then change the inside to a radical form and simplify it,
then take it to the power on the outside:

Lets look at one with a negative power:


Same step:

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