Simplifying Nested Square Roots

To find the value of the expression $\sqrt{(10 + \sqrt{(200 + \sqrt{(596 + \sqrt{841)))})}}$, we simplify the nested square roots starting from the innermost one.

Innermost Radical Calculation

• First, calculate $\sqrt{841}$. We know that $29^2 = 841$. So, $\sqrt{841} = 29$.

Stepwise Radical Evaluation

• Substitute this value back into the expression: $\sqrt{(10 + \sqrt{(200 + \sqrt{(596 + 29)})})}$ Calculate the sum inside the next square root: $596 + 29 = 625$.
• Now, find the square root of 625: $\sqrt{625}$. We know that $25^2 = 625$. So, $\sqrt{625} = 25$.
• Substitute this value back: $\sqrt{(10 + \sqrt{(200 + 25)})}$ Calculate the sum inside the next square root: $200 + 25 = 225$.
• Find the square root of 225: $\sqrt{225}$. We know that $15^2 = 225$. So, $\sqrt{225} = 15$.
• Substitute this value back: $\sqrt{(10 + 15)}$ Calculate the sum inside the final square root: $10 + 15 = 25$.
• Finally, calculate the outermost square root: $\sqrt{25} = 5$.

Final Value and Option Matching

The calculated value of the expression is 5.

Let's check the options:

• Option 1: $5\sqrt{3} \neq 5$
• Option 2: $5\sqrt{2} \neq 5$
• Option 3: $5\sqrt{5} \neq 5$
• Option 4: $5\sqrt{1} = 5 \times 1 = 5$. This matches our calculated value.

Therefore, the value of the expression is $5\sqrt{1}$.