Simplifying a radical with fraction and sum

Question

What are the steps to simplify

$(1+(\frac{1}{2}(x^3-\frac{1}{x^3})^2))^ \frac{1}{2}$

to

$\frac{1}{2}(x^3+\frac{1}{x^3})$ ?

Answer 1

Hint: Try using the identities $(a-b)^2=a^2-2ab+b^2$ and $(a+b)^2=a^2+2ab+b^2$ (which are actually the same identity).

Be careful: check whether the result holds for $x\lt0$.

I am assuming you meant $$ \left(1+\left(\frac12\left(x^3-\frac1{x^3}\right)\right)^2\right)^{1/2} $$

Answer 2

If what you have is $$ \sqrt{ 1 + \frac{1}{2}(x^3 - \frac{1}{x^3})^2} $$

$$ \sqrt{ 1 + \frac{(x^6 - 1)^2}{2x^6}} = \sqrt{ \frac{x^{12 } + 1}{2x^6}} = \sqrt{ \frac{1}{2}(x^6 + \frac{1}{x^6})}$$

So they are not equal.