Question

How do I factor (x^2 + 1)^1/2 +2(x^2 + 1)^1/2

Answer 1

`(x^(2) + 1)^{ (1)/(2)} + 2((x^(2) + 1)^{ -(1)/(2)}) \\\\ Take \ out \ (x^(2) + 1)^{ (1)/(2)} \ like \ so: \\\\ (x^(2) + 1)^{ (1)/(2)}(1 + 2(x^(2) + 1)^( -1)) \\\\ = (x^(2) + 1)^{ (1)/(2)}(1 + (2)/((x^(2) \ + \ 1))) \\\\ = (x^(2) + 1)^{ (1)/(2)}((2 \ + \ (x^(2) \ + \ 1))/((x^(2) \ + \ 1))) \\\\ = (x^(2) + 1)^{ (1)/(2)}(((x^(2) \ + \ 3))/((x^(2) \ + \ 1))) \\\\ = \frac{(x^(2) \ + \ 1)^{ (1)/(2)}(x^(2) \ + \ 3)}{(x^(2) \ + \ 1)} \\\\`
This is what I would think to do with such an expression;
It should be alright to leave it as it is on the last line as this should be its simplest form;
But, you could write it out as:

`\frac{(x^(2) \ + \ 3)}{(x^(2) \ + \ 1)^{ (1)/(2)}} \\\\ = \frac{(x^(2) \ + \ 3)}{ \sqrt{(x^(2) \ + \ 1)}}`