How do I get from [x(x-1)^-1/2]+[2(x-1)^-1/2] to (3x-2)/sqrt(x-1)?

Question

the original problem reads x(x-1)^-½ + 2(x-1)^-½. The answer is (3x-2)/(x-1)^½. How do I get from the original to the answer step-by-step?

Answer 1

To make things easier to read let y=(x-1)^-1/2=1/sqrt(x-1). The question becomes: xy+2y=y(x+2)=(x+2)/sqrt(x-1), not the answer you're expecting!

I think the expression should read: x(x-1)^-1/2+2(x-1)^1/2=x(x-1)^1/2+2(x-1)(x-1)^-1/2=xy+2(x-1)y=xy+(2x-2)y=y(x+2x-2)=y(3x-2)=(3x-2)/sqrt(x-1).

To prove the original was wrong, put x=5:

5*4^-1/2+2*4^-1/2=5/2+1=7/2; but (3x-2)/sqrt(x-1)=13/2, which doesn't match 7/2!

Now put x=5 in the revised expression: 5*4^-1/2+2*4^1/2=5/2+4=13/2, which does match!