Find the value of \( \sqrt{125 + \sqrt{344 + \sqrt{289}}} \).
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.
Important Questions from Simplification
Simplify the following expression.
\(\left(\frac{7}{16} \div \frac{1}{2}\:of\: \frac{1}{5}\right)\times \frac{4}{5}-\frac{1}{3}\times\frac{5}{8}\div \frac{1}{2}+\frac{3}{4}\)
The value of \(\left( {2\frac{6}{7}of4\frac{1}{5} \div \frac{2}{3}} \right) \times 5\frac{1}{9} \div \left( {\frac{3}{4} \times 2\frac{2}{3}of\frac{1}{2} \div \frac{1}{4}} \right)\) is:
The value of \(\left[ {\frac{4}{7}\rm \;of\;2\frac{4}{5} \times 1\frac{2}{3} - \left( {3\frac{1}{2} - 2\frac{1}{6}} \right)} \right] \div \left( {3\frac{1}{5} \div 4\frac{1}{2}\;\rm of\;\;5\frac{1}{3}} \right)\) is:
The value of \(\frac{{0.0203 \times 2.92}}{{0.7 \times 0.0365 \times 2.9}} \div \frac{{{{\left( {12.12} \right)}^2} - {{\left( {8.12} \right)}^2}}}{{{{\left( {0.25} \right)}^2} + \left( {0.25} \right)\left( {19.99} \right)}}\) is:
The value of 4 ÷ 12 of [3 ÷ 4 of {(4 - 2) × 6 ÷ 2}] - 2 × 6 ÷ 8 + 3 is:
