in this video we're going to focus on
simplifying exponents but let's go over
some basic properties what is X to the
fourth x x to the fifth what is that
equal to whenever you multiply by a
common base you need to add the
exponents 4 plus 5 is 9 X to the fourth
is basically multiplying 4 X variables
together and X to the fifth is
equivalent to multiplying five X
variables together so in total you're
multiplying 9 X variables together and
so that's why it's equal to X to the
ninth
now what about division let's say if you
divide by a common base what should you
do with the exponents you need to
subtract X it is 7 divided by X to the 3
it needs to subtract 7 by 3 and that's
going to be exit for X to the 7 is
equivalent to 7 X variables multiplied
to each other and X cubed is simply x
times X times X so we can cancel 3 X
variables which we'll leave behind for X
variables on top and so that's the
answer there's another one what is X to
the third raised to the fourth power
what should we do whenever we raise one
exponent to another in a situation like
this you need to multiply it's going to
be 3 times 4 which is 12 so X to the
third raised to the fourth power means
that you're multiplying 4 x cubed values
and each x cubed is basically three X
variables multiplied to each other
so if you count all the X variables what
we have is a total of 12 and so that's
why it's equal to X to the 12
now what about X raised to the 0 power
what is that equal to anything raised to
the 0 power is 1 so 4 to 0 is 1 now what
about X raised to the negative 3 what
should we do if we have negative
exponents if you have a negative
exponent all you could do is take the
variable move it to the bottom when you
do that the negative 3 will become
positive 3 and so it's 1 over X cubed
likewise let's say if you have 1 divided
by X raised to the negative 4 you can
move the X variable to the top any
exponent will change sign so it's going
to be X to the positive 4 over 1 or
simply X raised to the 4th power
that is the question theme what is the
value of negative 3 squared what do you
think that's equal to now what about
negative 3 squared within a parenthesis
negative 3 squared is basically negative
3 times negative 3 we have two of them
so this is equal to positive 9 now what
we have here is one negative and two
threes this 2 does not apply to the
negative sign so we only have one
negative sign so this is negative and
then 3 times 3 is 9 so the whole answer
is negative 9 so just keep in mind the
difference between the two you can
confirm this in a calculator if you type
in negative 3 squared just the way you
see it like this you should get negative
9 so tree typing like this in your
calculator it should give you positive 9
go ahead and simplify these problems
simplify the following expressions so
let's say this is the first one here's
the next one X cubed raised to the
negative 5 and also X to the seventh
power divided by X to the 12 go ahead
and work on these examples so when
multiplying by
when multiplying haben bases in to add
the exponents 4 plus negative 9 is
negative 5 and X raised to the negative
5 is equivalent to 1 over X to the 5th
so if you get any negative exponents
make it positive now here we have one
exponent raised to another so we need to
multiply 3 times negative 5 is negative
15 x to the negative 15 is 1 over X to
the positive 15 so this is the final
answer for that problem now for the last
one we need to subtract we're going to
take the top number and subtracted by
the bottom number 7 minus 12 is negative
5 and this is the same as 1 over X to
the 5 now let's talk about that example
X is 7th means that we have 7x variables
X to the 12
means that we have a total of 12 X
variables multiplied to each other so we
can cancel all 7 X variables on top and
seven on the bottom and notice that we
have five X variables left on the bottom
and that's why it's run over X to the 5
what is the value of 3x squared raised
to the third power go ahead and simplify
this expression so 3 is the same as 3 to
the first whenever you raise one
exponent to another you need to multiply
but we have 2 X 1 is here so we need to
distribute the 3 to the 1 and to the 2
so 3 times 1 is 3 and 2 times 3 is 6 so
this is 3 to the 3rd times X to the 6 so
let's drink to the 3rd 3 to the 3rd is 3
times 3 times 3 3 times 3 is 9 and 9
times 3 is 27 so the final answer is 27
times
x-rays to six power try this one
negative two okay that doesn't look like
a - let's do that again
negative two X cubed Y to the fourth
raised to the second power
feel free to pause the video and work on
that example so this is raised to the
first power and if we multiply by two
we're going to have negative two to the
second power and then three times two is
six so we're going to have X - 6 + 4
times 2 is 8 so this is going to be Y to
the negative 2 squared that's basically
negative 2 times negative 2 that's equal
to positive 4 so the answer is 4 X to
the sixth Y raised to the 8th power what
is the value of 5 X cubed x 4 X to the
seventh power so what should be doing
this example so first we need to
multiply 5 times 4 5 times 4 is 20 and
then we need to multiply X cubed times X
- 7 in which case we need to add 3 plus
7 is 10 so this is simply 20 X to the
10th so try this example 7 X to the 6
times 5 X to the 4th and also 4 X Q Y
squared times 7 X to the negative fourth
+ y to the 3rd so in the first example
we need to multiply 7 by 5 7 times 5 is
35 and then we got to multiply X - 6
times X to the 4th 6 plus 4 is 10 so
it's 35 X at a 10 now the next example
we're going to multiply 4 by 7 which is
28 and then X cubed times X and negative
fourth so we're going to add 3 plus
negative 4 which is a negative 1 and
then finally we got to multiply
y squared times y cube 2 plus 3 is 5 so
it's Y to the fifth power now we need to
get rid of the negative x one so let's
move the X variable to the bottom so
it's 28 y to the fifth divided by X and
so that's the answer what is negative 2
X cubed Y raised to the 0 power what is
that equal to anything raised to 0 power
keep in mind is 1 now what about
negative 4 X to the 0 power what is that
equal to you need to realize that this
is negative 4 times X raised to 0 power
this 0 only applies to X so this portion
highlighted in red that's equal to 1 so
it's negative 4 times 1 and that's
negative 4 so anything that's affected
by the 0 exponent will become a 1
anything outside of that is unaffected
try this problem what is 24x to the 7th
Y to the 3rd divided by 8 X to the 4th Y
to the negative 12 so in this case we'll
need to divide 24 divided by 8 is 3 X to
the 7th divided by X to the fourth is X
cubed 7 minus 4 is 3 and for the Y
variables we need to subtract 3 minus
negative 12 is the same as 3 plus 12 so
this becomes wife's insistence and so
that's the answer
what is 35 X cubed Y to the fifth
divided by 60 3 X to the negative fourth
Y to the seventh squared what is that
equal to
so we could distribute the two or we
could simplify first we could divide
first or distribute the exponent let's
divide first 35 is basically 5 times 7
and 63 is 9 times 7 now X to the 3rd
divided by X to the negative fourth
that's going to be 3 minus negative 4
which is 3 plus 4 and at 7 so we're
going to have X - 7 on top and then we
have 5 minus 7 which is negative 2 it's
a negative 2 on the top but if we bring
to the bottom it's going to be positive
2 on the bottom so let's go ahead and do
that and this is still raised to the
second power now the last thing we could
do is cancel the sudden let's get rid of
this stuff so we have 5 X to the 7th
divided by 9 y squared raised to the 2nd
power 5 squared is 25 and X to the 7th
raised to the second power it's going to
be X - 14 7 times 2 is 14 9 squared is
81 and 2 times 2 is 4
so this should be the final answer