The expression you're asking about is:

$\sqrt{x} + \sqrt{x} + \sqrt{x} + \cdots$

This suggests an infinite sum of square roots of $x$, but it could also represent a continuous operation involving radicals. To clarify the interpretation:

1. Sum of square roots: If you're asking about adding $\sqrt{x}$ repeatedly an infinite number of times, the sum would diverge unless $x = 0$.

2. Nested square roots: If you meant something like a nested radical, where:

$\sqrt{x + \sqrt{x + \sqrt{x + \cdots}}}$

Then, we need to solve this as an equation. Let this expression be equal to $y$:

$y = \sqrt{x + y}$

Now, square both sides to remove the square root:

$y^2 = x + y$

Rearrange the equation:

$y^2 - y - x = 0$

This is a quadratic equation in $y$, so we can solve it using the quadratic formula:

$y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-x)}}{2(1)}$ $y = \frac{1 \pm \sqrt{1 + 4x}}{2}$

Since $y$ must be positive, we take the positive root:

$y = \frac{1 + \sqrt{1 + 4x}}{2}$

Thus, the value of the infinite nested radical expression is:

$y = \frac{1 + \sqrt{1 + 4x}}{2}$

Would you like more details on this process?

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Here are 5 related questions for further exploration:

1. What happens if $x = 0$ in this nested radical equation?
2. Can this process be applied to different forms of nested radicals?
3. How does this solution compare to simpler radical equations like $\sqrt{x + \sqrt{x}}$?
4. What is the geometric interpretation of nested radicals?
5. How would the equation change if we added a constant inside the radical?

Tip: When solving equations with nested square roots, isolating the radical and squaring both sides is often the best strategy.