Question

Which expression is equivalent to the following complex fraction?

Answer 1

Option (3) is correct.

The given expression
`((x)/(x-3) )/((x^2)/(x^2-9) )` is equivalent to
`frac{x+3}{x}`

Explanation:

Given expression
`((x)/(x-3) )/((x^2)/(x^2-9) )`

We have to find an equivalent fraction to the given expression
`((x)/(x-3) )/((x^2)/(x^2-9) )` out of given options.

Consider the given expression
`((x)/(x-3) )/((x^2)/(x^2-9) )`

Divide fractions
`((a)/(b))/((c)/(d))=(a\cdot \:d)/(b\cdot \:c)`

We have,

`=(x\left(x^2-9\right))/(\left(x-3\right)x^2)`

Cancelling common factor x, we have

`=(\left(x^2-9\right))/(\left(x-3\right)x)`

Using algebraic identity
`(a^2-b^2)=(a+b)(a-b)`, we have,

Apply on
`x^2-9` we get,
`x^2-3^2=\left(x+3\right)\left(x-3\right)`

Substitute, we get,

`=(\left(x+3\right)\left(x-3\right))/(x\left(x-3\right))`

Cancelling out common factor (x-3) , we get

`=(x+3)/(x)`

Thus, the given expression
`((x)/(x-3) )/((x^2)/(x^2-9) )` is equivalent to
`frac{x+3}{x}`

Option (3) is correct.