Question

How do you simplify `\frac { 1- \frac { 4} { x ^ { 2} } } { \frac { 5} { x } + \frac { 10} { x ^ { 2} } }`?

Answer 1

Please see the explanation.

Explanation:

Multiply by 1 in the form `x^2/x^2`

`(1 - 4/x^2)/(5/x + 10/x^2)x^2/x^2`

Multiply numerators and denominators:

`(x^2 - 4x^2/x^2)/(5x^2/x + 10x^2/x^2)`

The denominators are canceled by common factors in the numerators:

`(x^2 - 4)/(5x + 10)`

The numerator is the difference of two squares; we know how that factors:

`((x - 2)(x + 2))/(5x + 10)`

Remove a factor of 5 from the denominator:

`((x - 2)(x + 2))/(5(x + 2))`

The common factor `x + 2` cancels:

`(x - 2)/5`

Simple enough, I think.