Question

Perform indicated operation?

`(2p)/(4p^2-1)/(6p^3)/(6p+3)`

Answer 1

`((2p)/(4p^2-1))/((6p^3)/(6p+3))` = `1/(p^2(2p-1))`

Explanation:

`((2p)/(4p^2-1))/((6p^3)/(6p+3))`

This would be the same as multiplying the numerator
`(2p)/(4p^2-1)`
with the inverse of the denominator
`(6p^3)/(6p+3)`, i.e. `(6p+3)/(6p^3)`

We, therefore, get:
`((2p)/(4p^2-1))/((6p^3)/(6p+3))`
= `((2p)/(4p^2-1))*((6p+3)/(6p^3))` = `(2p(6p+3))/(6p^3(4p^2-1))`
= `(3(2p+1))/(3p^2(4p^2-1))` = `((2p+1))/(p^2(4p^2-1))`

Now we need to remember that (a+b)(a-b)=a^2-b^2
Therefore (2x+1)(2x-1)=(4x^2-1)

We can, therefore, write:
`((2p+1))/(p^2(2p+1)(2p-1))=1/(p^2(2p-1))`