\( \sqrt{4} = 2\)

Algebra Level 4

Consider the function \( f: \mathbb{R} \to \mathbb{R} \) such that, for two defined constants \(a,b\), \[ f(x) = \dfrac{a^x - b^x}{a-b}.\]

We know that \( f(0) = 0, \; f(1) = 1, \; f(2) = \sqrt{2}, \; f(3) = \sqrt{3} \).

If \( f(4) = w_1\sqrt{w_2-\sqrt{w_3}} \), where \(w_1, w_2, w_3\) are all integers with \( w_3 \) is square-free, find \( w_1 + w_2 + w_3 \).

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