In this video we are going to talk about some of the properties of exponents, earlier in
the course we introduced the idea of an exponent so if you feel like you need to brush up on
those you are welcome to look back at those before starting this video.
Just as a very brief refresher what an exponent is is a number that we write in the upper
right hand corner of another term called a base and what exponent means is we are going
to multiply the base times itself repeatedly that number of times so for example if we
wrote 3 to the 4th power that means 4 is the exponent where in this case 3 is called the
base and we would have 3 times 3 times 3 times 3 that’s 4 times because that’s how large
the exponent is and so we can compute this number 3 times 3 is 9 9 times 3 is 27 and
27 times 3 is 81 so 3 raised to the 4th power will be 81 what we want to do in this video
is go a little deeper than that.
What happens if you have multiple terms that have exponents what sort of properties do
exponents have that can help us combine exponent and that sort of thing.
So let me just run through these properties real quick here for each of these I will just
show you a small example now the first one says if you have x to the m times x to the
n where the bases match one another this would be x to the m plus n you add the exponents
you do not multiply the exponents so for example if you had 2 to the 2nd maybe times 2 to the
3rd since these bases match one another we can say this is 2 to the 2 plus 3 which would
make 5 and 2 to the 5th we can actually compute 2 times 2 times 2 times 2 times 2 that’s
2 five times 2 times 2 is 4, 4 times 2 is 8, 8 times 2 is 16, 16 times 2 is 32 and so
we see why this property works actually now I won’t go into this much detail for the
rest of the properties but here’s 2 to the 2nd times 2 to the 3rd so if you add x to
the m and x to the n how many x’s would you have total you would have m plus n you
will have in this case 2 plus 3 which makes 5 of them total so that why we add the exponent
for this particular property.
Alright the next property what happens when you have an exponent and then an exponent
of the exponent of the term with the exponent so a power to a power what do we do in that
case well the property says that we will take x to the m times n so a power to a power we
are going to multiply let’s say we had y to the 3rd and all of that raised to the 5th
well this property says we have y to the 3 times 5 which would be 15 now make sure you
don’t confuse these a lot of students will mess this up and they will want to add 3 and
5 and that’s not correct if it’s a power to a power you multiply it like in this case
and we get y to the 15th the next property what happens if you have x to the m divided
by x to the n would just like when we had multiplication we added the exponents now
that we have division we subtract the exponents so for example let’s say we had 7 to the
8th divided 7 to the 6th well then based off of this property this should be 7 to the 8
minus 6 which will be 2 and 7 squared is 49 that’s exactly right.
We can also see this one graphically if we had seven times seven times seven times 7,
8 times and then divided by 7 times 7 times 7 six times 1 2 3 4 5 6 you see what’s going
to happen a lot of these 7 will cancel in fact every one of the 6 factors of seven will
cancel leaving you with just 7 squared in the numerator next property what if we have
x over y here the basis may not be the same like we had previously but x over y all to
the m’th power then you can distribute the power to the numerator and the denominator
and get x to the m divided whoops this should be y to the m there is no n sorry about that,
a little typo x to the m divided by y to the m so for example if we had 2 over x to the
3rd then just think about what this would be longways this is 2 over x times 2 over
x times 2 over x that’s 3 factors of 2 over x 2 times 2 times 2 is 2 cubed and x times
x times x would make x cubed so we could have skipped this middle step if we would have
noticed that 2 over x to the third would be 2 to the third divided by x to the third we
are just using this property here now let’s not leave it here though 2 to the third is
2 times 2 times 2 which is 8 so leave our final answer 8 over x to the third final answer.
Alright couple more and then we are done what if we have different bases x times y all raised
to the m’th power then this would be x to the m times y to the m the m distributes over
multiplication now as a quick little side note I want to make a caution because this
is such a common mistake that I see from students if you have x plus y to the m then absolutely
you cannot distribute the powers x to the m plus y to the m now you say Devin what’s
the difference they look the same to me this is addition and this is multiplication power
split over multiplication they do not split over addition so just be really careful about
that cause that’s a super super common mistake.
Alright let’s try an example let’s say we had the quantity 3z to the 3rd power well
if we didn’t know the property to cubed we would have to take 3z times 3z times 3z
but then if you reorder all the terms you get 3 to the 3rd times z to the third and
sure enough that would be this property x times y to the m equals x to the m times y
to the m now anytime you have a numerical value raised to a power if possible go ahead
and figure out what that is 3 time 3 is 9 9 times 3 makes 27 so we 27z to the 3rd and
the last one doesn’t really need much of an explanation x to the zero power is automatically
one so there’s not really much of an example I can show you there here I think I can try
to make up an example if you had 5 to the 3rd divided by 5 to the 3rd then using that
property that we saw on the last page here we should be able to subtract these exponents
then we get 5 to the 3 minus 3 which will make 5 to the zero I think we would all agree
with that.
Well what is 5 to the 3rd over 5 to the 3rd those are the same things so that’s going
to give you one.
So that’s how we have anything to the zero power is 1 that means if you express as a
fraction you have the same quantity in the numerator and the denominator to give you
an exponent of zero and when they cancel that will equal 1.
So anyways this is just some properties that we have for exponents I understand this is
probably not the most exciting video but we will try to use some of these properties in
some of our examples coming up shortly.