Question

How do you multiply and simplify `\frac { ( x ^ { 3} y ^ { 10} ) ( x ^ { 4} y ) } { ( x ^ { 8} y ^ { 2} ) ^ { 3} }`?

Answer 1

`[(x^3y^10)(x^4y)]/[(x^8y^2)^3]`

Remove the brackets in the numerator

`[x^7y^11]/[(x^8y^2)^3]`

Remove the brackets in the denominator

`[x^7y^11]/[x^24y^6]`

cancel by `x^7`

`y^11/[x^17y^6]`

cancel by `y^6`

`y^5/x^17`

Answer 2

`y^5/x^17color(white)(...)` Lot of explanation given

Explanation:

Given: `( (x^3y^(10))(x^4y) )/((x^8y^2)^3)`

`color(blue)("The numerator")`

Consider the example: `color(white)("d")2^2xx2^3color(white)("d")=color(white)("d")4xx8=32`
But this is the same as : `color(white)(".d")2^(2+3)color(white)(".d")=color(white)("ddd")2^5color(white)(".") = 32`

Applying this to the numerator `(x^3y^(10))(x^4y)`

Write as: `[x^3xx x^4][y^10xxy^1] = x^(3+4)xxy^(10+1)=color(red)(x^7y^11)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("The denominator")`

Consider the example:
`(2^4)^3color(white)("d")=color(white)("d")2^4xx2^4xx2^4color(white)("d")=2^(4+4+4)=color(white)("d")2^(4xx3)=2^12`

Applying this to the denominator `(x^8y^2)^3`

Write as: `x^(8xx3)xxy^(2xx3)= color(green)(x^24y^6)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Putting it all together")`

`( (x^3y^(10))(x^4y) )/((x^8y^2)^3) = (color(red)(x^7y^11))/(color(green)( x^24y^6))color(white)("d") = color(white)("ddd")x^7/x^24color(white)("dd")xxcolor(white)("ddd")y^11/y^6`

`color(white)("dddddddddddddddddd") =[x^7/x^7xx1/x^17]xx[y^6/y^6xxy^5/1]`

`color(white)("dddddddddddddddddd")=color(white)("ddd")[1/x^17]color(white)("dd")xxcolor(white)("dd")[y^5/1]`

`color(white)("dddddddddddddddddd")=color(white)("dddddddddd")y^5/x^17`