## Simplifying Exponents With Fractions, Variables, Negative Exponents, Multiplication & Division, Math

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

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