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

#[(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#

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)#
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#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)#
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#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#