Question

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

Answer 1

$(x^{2} + 1)^{ \frac{1}{2}} + 2((x^{2} + 1)^{ -\frac{1}{2}}) \\\\ Take \ out \ (x^{2} + 1)^{ \frac{1}{2}} \ like \ so: \\\\ (x^{2} + 1)^{ \frac{1}{2}}(1 + 2(x^{2} + 1)^{ -1}) \\\\ = (x^{2} + 1)^{ \frac{1}{2}}(1 + \frac{2}{(x^{2} \ + \ 1)}) \\\\ = (x^{2} + 1)^{ \frac{1}{2}}(\frac{2 \ + \ (x^{2} \ + \ 1)}{(x^{2} \ + \ 1)}) \\\\ = (x^{2} + 1)^{ \frac{1}{2}}(\frac{(x^{2} \ + \ 3)}{(x^{2} \ + \ 1)}) \\\\ = \frac{(x^{2} \ + \ 1)^{ \frac{1}{2}}(x^{2} \ + \ 3)}{(x^{2} \ + \ 1)}$
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:
$\frac{(x^{2} \ + \ 3)}{(x^{2} \ + \ 1)^{ \frac{1}{2}}} \\\\ = \frac{(x^{2} \ + \ 3)}{ \sqrt{(x^{2} \ + \ 1)}}$