How isolate y from $a=\sqrt{(y^2+x^2)^3} + \sqrt{(y^2-x^2)^3}$

Question

$$a=\sqrt{(y^2+x^2)^3} + \sqrt{(y^2-x^2)^3}$$

I need to isolate y in a family of such expressions. Can someone give me a guide to learn how deal with those... There's no further information or context. I've found a lot of information to work with powers but it's not easy to find information about radicals. Maybe there's some kind of algorithm or method to go ahead with these expressions, but I've been unable to find any.

Answer 1

In some answers from yesterday's post, the main takeaway is not to be afraid of the gangliness of the expression. Additional tips to help:

• Square the right hand side, using the rule, $(x+y)^2 = x^2 + 2xy + y^2$
• Treat square roots as half-powers: $\sqrt{x} = x^{1/2}$
• Remember that products of powers with the same exponent combine their bases through multiplication, powers of same bases add, and powers-of-powers multiply: $$x^j\cdot y^j = (xy)^j \\ x^j\cdot x^k = x^{j+k} \\ (x^j)^k = x^{jk}$$

Try with these rules, first. There is value in working these things out by hand. Later, you can find online solvers that will help verify the answer.