Totally Radical

Given problem is : $\sqrt{5\sqrt{5\sqrt{5\sqrt{.....}}}}$

Let this be considered as $x$ . So ------>

$\sqrt{5\sqrt{5\sqrt{5\sqrt{.....}}}}=x$

On squaring both sides -----

$\implies 5\times \sqrt{5\sqrt{5\sqrt{5\sqrt{.....}}}}=x^2$

$\implies 5x=x^2$ [substituting original equation here]

$\implies x^2-5x=0$

$\implies x(x-5)=0$

$\implies x=\boxed{0} \text{ or } x-5=0 \implies x=\boxed{5}$

Now, we can clearly see that $x$ should be more than $0$ , i.e., $x\gt 0$ .

So, the required possible value of $x=\boxed{5}$

x=string ----------------(1) x^2=5string from (1) x^2=5x x=5

Let's call the given the "[string]" because typing the given is extremely time consuming and it'll be more confusing if I typed it.Trust me.

1. [string] = x
2. 5[string] = x^2

Since "[string] = x", we can substitute the "[string]" on the left side of the equation with "x". That leaves us with:

1. 5x = x^2
2. (5x)/(x) = (x^2)/(x)
3. 5 = x

The answer is 5