There is another property that deals with exponents, it is

 

 

 

Notice that the negative sign in front of the exponent has absolutely nothing to do with the result being negative or positive.

 

Example

Simplify

 

 

Applying the property.

 

 

 

 

Example

 

Simplify

 

Applying the properties of exponents.

 

 

 

Answer!

 

 

 

 

 

Using the property of negative exponents, another rule is

 

 

 

This entire idea of negative exponents is best understood thinking about it like this, “ If you move a number or variable from one side of a division property to another (side refers to top or bottom), it changes the sign of its exponent.” This is very important because there cannot be an answer with a negative exponent.

 

The first step in solving any kind of problem with negative exponents is to first make all the exponents positive. Remember to make a negative exponent positive move it “across the line” or from the numerator (top) to the denominator (bottom) or vice versa.

 

 

 

 

Example

Simplify

 

The first thing is to move the to the top and make it positive.

 

 

 

Now apply the properties of exponents 3+5 = 8.

 

 

 

Answer!

 

 

 

 

 

Example

Simplify

 

 

 

The first thing is to make all negative exponents positive. Move the to the bottom and the to the top.

 

 

 

 

Combine like bases (the x’s on top and the z’s on bottom).

 

 

 

 

Get all the y’s on either the top or bottom.

 

Note- To get the y’s all together, ask this question, “ Which side has the most y’s (top or bottom)?” Look at their exponents and realize there are 2 y’s on the top and 5 y’s on the bottom. Since there are more on bottom, the final answer will have the y’s on bottom. To find out how many, just subtract the exponents. 5-2 = 3.

 

 

 

This is almost the final answer. The last thing to do is to see if the numbers will simplify, they will since 3 goes into both of them, = .

 

 

Answer!

 

 

 

 

Example

Simplify

 

 

Get rid of the parentheses and raise everything in the parentheses to the 4^th power. Use the properties of exponents.

 

 

 

Notice the * sign between the 2 and 5. This is done so not to confuse oneself by thinking it is 25 instead of 2*5⁴. Make all the exponents positive by moving them to the other side.

 

 

Simplify the a’s. Since there are more on top, leave them on top and 8 –3 = 5. Also simplify the 5’s. Treat them just as if they were letters. There are more 5’s on top and 4-2=2.

 

 

 

5²= 25. And 2 * 25 = 50.

 

 

 

 

Answer!