Fractional Expression

[Gragadoodle]

There is supposed to be a really easy way to do this. How?

 

[pka]

There is supposed to be a really easy way to do this. How?

\(\displaystyle \\\left( {1 - \dfrac{1}{x}} \right)\left( {1 - \dfrac{1}{{x + 1}}} \right)\left( {1 - \dfrac{1}{{x + 2}}} \right)\left( {1 - \dfrac{1}{{x + 3}}} \right) =\\ \left( {\dfrac{{x - 1}}{x}} \right)\left( {\dfrac{x}{{x + 1}}} \right)\left( {\dfrac{{x + 1}}{{x + 2}}} \right)\left( {\dfrac{{x + 2}}{{x + 3}}} \right)\)

 

[Gragadoodle]

\(\displaystyle \\ \left( {\dfrac{{x - 1}}{x}} \right)\left( {\dfrac{x}{{x + 1}}} \right)\left( {\dfrac{{x + 1}}{{x + 2}}} \right)\left( {\dfrac{{x + 2}}{{x + 3}}} \right)\)

I got this far, but I didn't know what to do next.

 

[DrPhil]

\(\displaystyle \\ \left( {\dfrac{{x - 1}}{x}} \right)\left( {\dfrac{x}{{x + 1}}} \right)\left( {\dfrac{{x + 1}}{{x + 2}}} \right)\left( {\dfrac{{x + 2}}{{x + 3}}} \right)\)

I got this far, but I didn't know what to do next.

any common factor in both the numerator and denominator can be canceled out.

 

[Gragadoodle]

I got it; I feel really stupid for not noticing that