Question

Evaluate:

$\sqrt{\left(2 + \sqrt{\left(39 + \sqrt{\left(97 + \sqrt{(5 + \sqrt{(16)})}\right)}\right)}\right)}$

• A. 2
• B. 4
• C. 3
• D. 7

Answer 1

Answer:

C. 3

The evaluated value of this nested square root expression is 3.

$\sqrt{2 + \sqrt{39 + \sqrt{97 + \sqrt{5 + \sqrt{16}}}}}$

1. Step 1: Solve the innermost square root $\sqrt{16}$:

   • $\sqrt{16} = 4$

   • The expression becomes: $\sqrt{2 + \sqrt{39 + \sqrt{97 + \sqrt{5 + 4}}}}$

2. Step 2: Add and solve the next square root $\sqrt{5 + 4}$:

   • $\sqrt{9} = 3$

   • The expression becomes: $\sqrt{2 + \sqrt{39 + \sqrt{97 + 3}}}$

3. Step 3: Add and solve the next square root $\sqrt{97 + 3}$:

   • $\sqrt{100} = 10$

   • The expression becomes: $\sqrt{2 + \sqrt{39 + 10}}$

4. Step 4: Add and solve the next square root $\sqrt{39 + 10}$:

   • $\sqrt{49} = 7$

   • The expression becomes: $\sqrt{2 + 7}$

5. Step 5: Solve the final outer square root $\sqrt{2 + 7}$:

   • $\sqrt{9} = \mathbf{3}$