FRACTIONAL EXPONENTS
OBJECTIVE
Review
What is an exponential expression?
An exponential expression is a representation of a number being multiplied, multiple times. It has two components, a base and the exponent.
What is the base of an exponent?
The number that is being multiplied multiple times.
What is the exponent part?
The number of times the number is being multiplied by itself.
How do we multiply 2 numbers with exponents, that share the same base?
We add the exponents.
SO, HOW DO WE MULTIPLY EXPONENTS?
BUT WHAT ABOUT WHEN WE DON’T HAVE THE SAME BASE?
SO WHAT ABOUT DIVISION?
-
And this is the property of dividing exponents:
When dividing two numbers of the same base, simply subtract the exponents together to get the answer.
THE PROPERTY OF DIVIDING EXPONENTS
A FEW THINGS TO NOTE:
Negative Exponents
So, why is a negative exponent a fraction?
Those freaking negatives!!!!
But this makes sense right? Since we know that there really is no such thing as subtraction, just the addition of negative numbers.
In the same way, there’s really no such thing as division, just the multiplication of numbers with negative exponents.
So what are fractions?�
So, to start off with, let’s actually look at what we know about fractions (other than we hate them).
So, let’s start with an easier one, let’s say we have one whole pizza:
Well, we know that means that we have a whole pizza, and we are dividing that whole pizza into two pieces.
So:
Now, what does this mean?
Well, it means that we broke the pizza into two pieces, that add to one whole piece.
Or in other words:
+ = 1
It’s the same thing with exponents
Again, when we break a whole number into a fraction evenly (1/2, ¼, etc.) , we are dividing that whole number into pieces that add up into 1.
So, when it comes to fractional exponents, it’s essentially the same thing except:
We are dividing the whole number into pieces that multiply into the base.
Here’s what I mean:
And they ask us to solve.
What they are really asking is, if we break 100 into two pieces that are multiplied together, what would they be?
Or in other words:
___ * ___ = 100
10 10
And that is essentially it!
So, when we have a fractional exponent, we’re looking for a number that multiplies together however many times the denominator says.
Again, this sounds way more complicated than it is.
Example 1:
4 4 4 4
So, the answer here is 2!
Example 2:
2 2 2
So, the answer here is 4!
Example 3:
So, the answer here is 3!
Example 4:
3 3 3
But, is 5 the only answer here?
What else can we put?
Let’s re-look at this.
___ * ___ = 25
5 5
Here’s a hint:
+
-5 -5
And there is the rule
When the denominator of the number is even, then we need to make sure to add the negative as well.
If the denominator is odd, then the answer’s sign will be whatever sign the question was. �Sounds complicated, but here’s an example to show you what I mean:
Example 1:
But remember, we have another answer as well:
___ * ___= 49
Example 2:
Remember, we have an odd denominator, so our answer will take on the same sign as the base.
So, in this case, since our base is -8, then our answer must be negative as well.
So our answer is -2
7 7
-7 -7
So our answer would be 7, -7
-2 -2 -2
Example 3:
Remember, we have an odd denominator, so our answer will take on the same sign as the base.
So, in this case, since our base is -625, then our answer must be negative as well.
So our answer is -5
Example 4:
Again, we have an odd denominator, so our answer will take on the same sign as the base.
So, in this case, since our base is 8, then our answer must be positive as well.
So our answer is 2
-5 -5 -5 -5 -5
2 2 2