$\sqrt{59 + \sqrt{16 + \sqrt{81}}} = \text{?}$

Answer: 8

Step-by-step solution

Step 1: Simplify innermost square root

We begin simplifying nested radicals by evaluating the innermost square root first. Since $9^2 = 81$ , the principal square root of $81$ is $9$ .

Step 2: Evaluate the intermediate radical

Substitute $9$ in place of $\sqrt{81}$ . This simplifies the second radical to $\sqrt{16 + 9} = \sqrt{25}$ , which evaluates to $5$ .

Step 3: Evaluate the outermost square root

Substitute $5$ into the outer radical to obtain $\sqrt{59 + 5} = \sqrt{64}$ . Since $8^2 = 64$ , the final value is $8$ .

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