Question

$\sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5} } } } =$
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Answer 1

Question:-

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

Answer:-

$\sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5} } } } =$

= ${5}^{ \frac{1}{2} } \times {5}^{ \frac{1}{4} } \times {5}^{ \frac{1}{8} } \times {5}^{16}$

{As ⁿ√a^m = a^(m/n).}

= ${5}^{ \frac{8 + 4 + 2 + 1}{16} }$

= ${5}^{ \frac{15}{16} }$ (ans.)

More:-

If you had to evaluate

$\sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5 \sqrt{... \infty } } } } } ,$

Let, $\sqrt{5 \sqrt{5 \sqrt{5 \sqrt{... \infty } } } } = x$

So,

$\sqrt{5.x} = x$

=> $x^2 = 5x$ (by SBS)

=> $x = 5$.

So then, the value would have become 5.