Question

How do you multiply and simplify `\frac { 3p ^ { 4} } { 5q ^ { 5} ( r - 5) ^ { 3} } \cdot \frac { 4l q ^ { 2} ( r - 5) } { 21p ^ { 3} }`?

Answer 1

`(4pl)/(35q^3(r-5)^2)`

Explanation:

`(color(green)3p^color(red)4)/(5q^5(r-5)^3)*(4lq^2(r-5))/(color(green)21p^color(red)3)`

Subtract the smaller exponent of a base from both bases, because of the division, and simplify `3` and `21` by dividing them by `3`

`(cancel3p^(4-3))/(5q^5(r-5)^3)*(4lq^2(r-5))/(7cancel21cancel(p^(3-3)))`

Anything to the power of zero can be cancelled

`(r-5)` is the same as `(r-5)^1`

`=p/(5q^5(r-5)^color(red)3)*(4lq^2(r-5)^color(red)1)/7`

`=p/(5q^5(r-5)^(color(red)3-1))*(4lq^2cancel((r-5)^(color(red)1-1)))/7`

`=p/(5q^color(red)5(r-5)^2)*(4lq^color(red)2)/7`

`=p/(5q^(color(red)5-2)(r-5)^2)*(4lcancel(q^(color(red)2-2)))/7`

`=p/(5q^3(r-5)^2)*(4l)/7`

Now there is no other simplification, we will multiply these two expressions together

`=(p*4l)/(color(green)5q^3(r-5)^2*color(green)7)`

`=(4pl)/(35q^3(r-5)^2)`

After a final check, this is the most simplified form.