The value of $\sqrt{ \left(42 + \sqrt{ \left(46 + \sqrt{ \left(3 +\sqrt{ \left(34+ \sqrt{(4)}\right)} \right) }\right)}\right)}$ is:
Nested Square Roots Evaluation
This problem requires simplifying a mathematical expression with multiple nested square roots. We solve it step-by-step, starting from the innermost root and working outwards.
Radical Expression Step-by-Step Solution
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Evaluate the innermost square root: $\sqrt{(4)}$.
Calculation: $ \sqrt{(4)} = 2 $
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Substitute this result into the expression $34 + \sqrt{(4)}$.
Calculation: $34 + 2 = 36$.
Evaluate the square root: $ \sqrt{36} = 6 $
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Substitute this result into the expression $3 + \sqrt{(34+ \sqrt{(4)})}$.
Calculation: $3 + 6 = 9$.
Evaluate the square root: $ \sqrt{9} = 3 $
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Substitute this result into the expression $46 + \sqrt{(3 + \sqrt{(34+ \sqrt{(4)})})}$.
Calculation: $46 + 3 = 49$.
Evaluate the square root: $ \sqrt{49} = 7 $
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Substitute this result into the final expression $42 + \sqrt{(46 + \sqrt{(3 + \sqrt{(34+ \sqrt{(4)})})})}$.
Calculation: $42 + 7 = 49$.
Evaluate the final square root: $ \sqrt{49} = 7 $
Final Value Determination
The step-by-step simplification confirms that the value of the given nested square root expression is 7.
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:
