$\"\"$

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The number is $\"\"$ .

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Consider $\"\"$ .

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$\"\"$ .

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Newtons approximation formula : $\"\"$ .

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Consider $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$ .

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From the newtons method, $\"\"$ .

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We have $\"\"$ and $\"\"$ is close to the $\"\"$ .

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So consider $\"\"$ .

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Substitutte $\"\"$ in $\"\"$ .

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$\"\"$

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Substitutte $\"\"$ in above expression.

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$\"\"$

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Repeat the process to get the value of the root, correct to eight decimal places are same.

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$\"\"$

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Consider $\"\"$ .

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Substitutte $\"\"$ in above expression.

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$\"\"$ .

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Substitutte $\"\"$ in above expression.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Hence $\"\"$ .

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Approximation of the number $\"\"$ is $\"\"$ .

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$\"\"$

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Approximation of the number $\"\"$ is $\"\"$ .