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Rules of Exponents - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Since we often see exponents throughout all math courses, it is important to understand the rules of exponents. We need to understand how to distribute, add, multiply and divide exponents in order to simplify expressions or manipulate equations that have exponents. The rules of exponents, like those involving multiplication of terms, are important to learn and will be used throughout Algebra I and II and Calculus.

So as you know an exponent is the little number above a base, so if you have like 3 squared, the 2, 3 the second is your exponent. And basically all it's telling you is to multiply the base times itself that many times. So 3 to the fourth is just 3 times 3, times 3, times 3. Or 3 times 3 is 9 times 3 is 27, times 3 is 81. Okay and with exponents come a long a bunch of rules that we're going to go through one at time. Okay so let's start over here, our first rule is that, if our bases are the same we're multiplying 2 terms of exponents and our bases are the same we can just add our exponents together. So let's say I have 3 squared times 3 to the third. Basically I have 2 times 2 here, 2 times 2 times 2 here in that sense where I really have is 5 2's and this I just going to equal 2 to the fifth.
The other way of doing that is just adding 2 and 3. Similarly if we are dividing okay, if have the same base and have our exponents we then just subtract. And this one actually depends on which number is bigger, so if we have 3 to the fourth over just say 3 there's really a imaginary one right here. So only have 3 to the 4 minus 1, this is just going to be 3 to the third okay. If our power in the denominator is larger we're just going to be left with a power in the bottom okay. So another example of this we'd say like 5 squared over 5 to the fourth. We still go 5 to the numerator exponent minus denominator exponent but what we end up with is 5 to the negative second. We'll talk about it in a minute but basically what that negative exponent means is it's left in the denominator.
Okay so really the easiest way to look at this is first just look at your exponent see which one is bigger your numerator or denominator and that's going to be where your term ends up being. Okay another rule is anything to the zero power is equal to 1, easy enough 4 to the zero 1, 822 to the zero 1, okay anything to the zero of power is just going to be 1. This one is already touches back to what I talked about over with the division one. Anything to a negative exponent is basically going to put that in the denominator of a fraction so if we have 4 to the negative third this is just equal to 1 over 4 to the third okay. Basically it takes the same term, the same number just moves it down to the denominator. Common mistake is people like to think that okay this 3 can come out in front and makes the entire thing negative. No it just moves a term that's normally in the top down to the bottom.
Likewise if we have a negative exponent in the denominator, it's just going to move it up to the top okay. So anytime you see negative exponent really all that does is it takes something in the numerator moves it to the denominator, something in the denominator moves it up to the numerator okay. Moving on if we have a fraction to a power that power gets distributed into everything okay. So what we really end up doing is taking the m and putting it to both the a and the b ending up with a over m, b over m okay. So alright a to the m, b to the m, so example if we have 2 third squared what we really end up with is 2 squared over 3 squared which is the same thing as 4 over 9. That pen is about dead.
Alright same idea for multiplication, if we are multiplying inside of parenthesis and we have to a power this power can be distributed in. So say we have 3x to the fifth that 5 can get distributed in to both the 3 and the x giving us 3 to the fifth x to the fifth okay. Couple more to go, anytime we have an exponent, a term to an exponent to an exponent we just multiply these out okay. So what we end up with is a to the m to the n just turns up to a to the m times n. Example of this say we have 2 cubes to the fourth okay, all we have to do is say 2 to the 3 times 4 which turns into 2 to the twelfth okay. You could just write this out if you wanted to this is really 2 cubed, times 2 cubed, times 2 cubed, times 2 cubed each one has 3 so in essence we have 12 twos but this is a good shortcut to keep in mind.
And our last rule that we're going to talk about is anything, a fraction to a negative power okay, this is basically just going to be the denominator to this power over the numerator to the power. Reason being is we can go back to this statement up here and basically distribute this negative and n, and what would end up with then is a to the negative n over b to the negative n. Remembering our rules and negatives from over there a to the negative n is just going to be a to the n in the denominator b to the negative n is just going to be a b to the n in the numerator. So really all we did is distribute this term n and then the negative flips our fraction okay. So a lot of different rules they do take sometime to get used to and remembering how they work but the main thing is remember if you are multiplying your bases you add your exponent if you are taking a power to a power you multiply and anytime you see a negative exponent that is just going to flip where your term is. If it was in the numerator it goes to the denominator, denominator it goes to the numerator and if you have a fraction the negative exponent is just going to flip your fraction altogether.
Okay so a lot to remember but take sometime play around with them a little bit and I'm sure you'll be fine.

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