Question

$\sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7} } } } }$
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Answer 1

$\sf\sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7 } } } } } \\ \\ \sf \: as \: \: \: \sqrt{7 } = {7}^{ \frac{1}{2}} \\ \\ \sf\sqrt{7 \sqrt{7 \sqrt{7 \sqrt{7 \times {7}^{ \frac{1}{2} } } } } } \\ \\ \\ \sf \: {7}^{ \frac{1}{2} } \times {7}^{ \frac{1}{2} } \times {7}^{ \frac{1}{2} } \times {7}^{ \frac{1}{2} } \times {7}^{ \frac{1}{2} } \\ \\ \sf \: as \: \: base \: \: is \: \: same \ \\ \sf so \: \: powers \: \: will \: \: add. \\ \\ \sf {7}^{ (\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2}) } \\ \\ \boxed{ \red{\sf \: {7}^{ \frac{5}{2} } = {( \sqrt{7} )}^{5} = 49 \sqrt{7} = 129.64}}$