Solving Nested Square Roots Step-by-Step

To find the value of the expression \( \sqrt{125 + \sqrt{344 + \sqrt{289}}} \), we need to simplify it by starting from the innermost square root and working outwards.

Step 1: Simplify the Innermost Square Root

First, calculate the square root of 289:

• \( \sqrt{289} \)
• We know that \( 17 \times 17 = 289 \).
• Therefore, \( \sqrt{289} = 17 \).

Step 2: Substitute and Simplify the Next Layer

Now substitute the result back into the expression:

• \( \sqrt{125 + \sqrt{344 + 17}} \)
• Add the numbers inside the second square root: \( 344 + 17 = 361 \).
• The expression becomes \( \sqrt{125 + \sqrt{361}} \).

Step 3: Calculate the Second Square Root

Next, calculate the square root of 361:

• \( \sqrt{361} \)
• We know that \( 19 \times 19 = 361 \).
• Therefore, \( \sqrt{361} = 19 \).

Step 4: Substitute and Simplify the Final Layer

Substitute this result back into the expression:

• \( \sqrt{125 + 19} \)
• Add the numbers inside the final square root: \( 125 + 19 = 144 \).
• The expression simplifies to \( \sqrt{144} \).

Step 5: Calculate the Final Square Root

Finally, calculate the square root of 144:

• \( \sqrt{144} \)
• We know that \( 12 \times 12 = 144 \).
• Therefore, \( \sqrt{144} = 12 \).

Final Answer

The value of the expression \( \sqrt{125 + \sqrt{344 + \sqrt{289}}} \) is 12.