Final answer:
To solve the given equation by isolating one radical and squaring both sides, it requires isolating \( \sqrt{x} \), squaring both sides, and then further manipulation to ultimately remove the radical.
Step-by-step explanation:
The equation \( \sqrt{x+6} + \sqrt{x} = 8 \) can be solved by isolating one of the radical terms and then squaring both sides of the equation. Isolating \( \sqrt{x} \) and squaring would give us:
\( (\sqrt{x+6} - \sqrt{x})^2 = (8 - \sqrt{x})^2 \)
Expanding both sides and simplifying would result in:
\( x+6 - 2\sqrt{x+6}\sqrt{x} + x = 64 - 16\sqrt{x} + x \).
As the equation contains a radical term \( \sqrt{x} \), further manipulation is needed to isolate and square the term again to remove the radical. This process can be extensive and requires careful simplification and the use of algebraic identities.