Discussion

Question:

Simplify: $\sqrt{\sqrt[3]{\sqrt[4]{\sqrt[5]{\sqrt[6]{8^{240}}}}}}$

Explanation:

Let x = $\sqrt{\sqrt[3]{\sqrt[4]{\sqrt[5]{\sqrt[6]{8^{240}}}}}}$

⇒ x = ${({({({({(8^{240})}^{\frac{1}{6}})}^{\frac{1}{5}})}^{\frac{1}{4}})}^{\frac{1}{3}})}^{\frac{1}{2}}$

⇒ x = $8^{240 \times \frac{1}{6} \times \frac{1}{5} \times \frac{1}{4} \times \frac{1}{3} \times \frac{1}{2}}$

⇒ x = $8^{\frac{240}{720}}$

⇒ x = $8^{\frac{1}{3}}$

⇒ x = ${(2^{3})}^{\frac{1}{3}}$

⇒ x = $2^{3 \times \frac{1}{3}}$ = 2

Hence, option (a).

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