Question

√6√6√6√6................
if the question is not understood see the attacment

Answer 1

let x be the √6√6√6···
                              x=√6√6√6√6√6√6········
                             x²={√6√6√6√6√6....}²
                             x²=6√6√6√6√6√6·····
                              x²=6×x
                              x²/x=6
                              x=6

Answer 2

$Let\ x =6 \\ \\ y = \sqrt{x \sqrt{x \sqrt{x \sqrt{x....} } } } \\ \\ y = \sqrt{x\ \ \ ( \sqrt{x \sqrt{x \sqrt{x \sqrt{x....} } } })} \\ \\ y = \sqrt{x y} \\ \\ y^2 = x y \\ \\ y = x = 6$
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We can also do this in another way adding the powers of 6.

$y = x^{\frac{1}{2}} * x^{\frac{1}{4}} * x^{\frac{1}{8}} * x^{\frac{1}{16}} * x^{\frac{1}{32}} * ...... \\ \\ It\ is \ an\ infinite \ geometric\ series\ with\ a=1/2\ and\ ratio=1/2 \\ \\ y = x^{ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} ....} \\ \\ y = x^{\frac{1}{2}(\frac{1}{1-\frac{1}{2}})} \\ \\ y = x^{\frac{1}{2}*\frac{1}{\frac{1}{2}}} \\ \\ y = x = 6$