Question

How do you simplify `\frac { - ( \frac { \pi R ^ { 2} } { 16} ) ( \frac { R } { 4} ) } { \frac { 16\pi R ^ { 3} - \pi t R ^ { 2} } { 16} }`?

Answer 1

See a solution process below:

Explanation:

First, multiply the terms in the top part of the fraction:

`(-((piR^2)/16)(R/4))/((16piR^3 - pitR^2)/16) =>`

`((-piR^2 * R)/(16 * 4))/((16piR^3 - pitR^2)/16) =>`

`((-piR^3)/64)/((16piR^3 - pitR^2)/16)`

Now, we can use this rule for dividing fractions to simplify the expression:

`(color(red)(a)/color(blue)(b))/(color(green)(c)/color(purple)(d)) = (color(red)(a) xx color(purple)(d))/(color(blue)(b) xx color(green)(c))`

`(color(red)(-piR^3)/color(blue)(64))/(color(green)(16piR^3 - pitR^2)/color(purple)(16)) =>`

`(color(red)(-piR^3) xx color(purple)(16))/(color(blue)(64) xx color(green)((16piR^3 - pitR^2))) =>`

`(color(red)(-piR^3) xx color(blue)(cancel(color(purple)(16))))/(color(purple)(cancel(color(blue)(64)))4 xx color(green)((16piR^3 - pitR^2))) =>`

`color(red)(-piR^3)/(4color(green)((16piR^3 - pitR^2))) =>`

`color(red)(-piR^3)/(color(green)((64piR^3 - 4pitR^2))) =>`

`color(red)(piR^2 * -R)/((piR^2 * 64R) - (piR^2 * 4t)) =>`

`color(red)(piR^2 * -R)/(piR^2(64R - 4t)) =>`

`color(red)(color(black)(cancel(color(red)(piR^2))) * -R)/(color(red)(cancel(color(black)(piR^2)))(64R - 4t)) =>`

`(-R)/(64R - 4t) =>`

`(-1)/-1 xx (-R)/(64R - 4t) =>`

`(-1 xx -R)/(-1(64R - 4t)) =>`

`R/(-64R + 4t) =>`

`R/(4t - 64R) =>`