Question

Find the value of \[\sqrt 7 \] up to six decimal places by long division method.

Answer 1

Hint:
Here, we will find the value of an irrational number up to six decimal places by long division method. We will find the square root of the given number by long division method and then correct the value up to six decimal places. Thus, the value of an irrational number up to six decimal places is the required answer.

Complete step by step solution:
We are given an irrational number \[\sqrt 7 \].
Now, we will find the square root of 7 by long division method.
\[ \Rightarrow \] \[\begin{array}{l}2\mathop{\left){\vphantom{1\begin{array}{l}700000000\\\underline 4 \end{array}}}\right.
\!\!\!\!\overline{\,\,\,{\begin{array}{l}700000000\\\underline 4 \end{array}}}}
\limits^{\displaystyle\,\,\, {2.64575131}}\\46\left){\vphantom{1\begin{array}{l}300\\\underline {276} \end{array}}}\right.
\!\!\!\!\overline{\,\,\,{\begin{array}{l}300\\\underline {276} \end{array}}}\\524\left){\vphantom{1\begin{array}{l}2400\\\underline {2096} \end{array}}}\right.
\!\!\!\!\overline{\,\,\,{\begin{array}{l}2400\\\underline {2096} \end{array}}}\\5285\left){\vphantom{1\begin{array}{l}30400\\\underline {26425} \end{array}}}\right.
\!\!\!\!\overline{\,\,\,{\begin{array}{l}30400\\\underline {26425} \end{array}}}\\52907\left){\vphantom{1\begin{array}{l}397500\\\underline {370349} \end{array}}}\right.
\!\!\!\!\overline{\,\,\,{\begin{array}{l}397500\\\underline {370349} \end{array}}}\\529145\left){\vphantom{1\begin{array}{l}2715100\\\underline {2645725} \end{array}}}\right.
\!\!\!\!\overline{\,\,\,{\begin{array}{l}2715100\\\underline {2645725} \end{array}}}\\5291501\left){\vphantom{1\begin{array}{l}6937500\\\underline {5291501} \end{array}}}\right.
\!\!\!\!\overline{\,\,\,{\begin{array}{l}6937500\\\underline {5291501} \end{array}}}\\52915023\left){\vphantom{1\begin{array}{l}164599900\\\underline {158745069} \end{array}}}\right.
\!\!\!\!\overline{\,\,\,{\begin{array}{l}164599900\\\underline {158745069} \end{array}}}\\529150261\left){\vphantom{1\begin{array}{l}585483100\\\underline {529150261} \end{array}}}\right.
\!\!\!\!\overline{\,\,\,{\begin{array}{l}585483100\\\underline {529150261} \end{array}}}\end{array}\]
Thus, the square root of 7 is \[\sqrt 7 \simeq 2.64575131...\]
Now, we will find the square root of 7 and correct up to six decimal places.
\[ \Rightarrow \sqrt 7 \simeq 2.645751\]

Therefore, the value of \[\sqrt 7 \] up to six decimal places by long division method is \[2.645751\].

Additional Information:
We should know the rules for a number to be a perfect square which is the easiest method to find the perfect square. If the unit digit is 1, 4, 5, 6, 9 and 0, then the number must be a perfect square. If the unit digit is 5, then its ten’s digit is always 2. If the unit digit is 6, then its tens digit is always odd. If the unit digit is 1, 4, 5, 9, 0, then the tens digit is always even.

Note:
We know that a number multiplied by itself is known as a square number or perfect square. We can also find the square root of the number by prime factorization method, Repeated Subtraction method, number line method and average method. Prime factorization method and Repeated Subtraction method is applicable only when the given number is a perfect square number. We know that the given number is not a Perfect square number.We should remember that the last digit of the divisor and the quotient remains the same in the long division method.