Find the value of $\sqrt{3}$ up to six decimals.

Question

Vedantu Content Team · Accepted Answer

Hint: We first use the division of the square root process to find the square root of 3. We use the concept for both integer and decimal to find the value of $\sqrt{3}$ till the 6-digit place after decimal.Complete step-by-step answer:We first have to find and check if the number 3 is a square number or not. 3 is not a square number and that’s why it can’t be broken in factorised from.Steps to find square root using square root:Step 1:Write down the number whose square root is to be found and place a bar over the pairs of two numbers from unit’s place. We get $3=3.0000000000$ $$\overline{3}.\overline{00}\overline{00}\overline{00}\overline{00}$$Step 2:For divisor, take the largest number whose square is less than or equal to the first pair of number. Here, first pair is 3. So, 1 is the largest number whose square is less than 3. So, we will divide by 1.$$\begin{align}  & 1 \\  & 1\left| \!{\overline {\,  \begin{align}  & \overline{3}.\overline{00}\overline{00}\overline{00}\overline{00} \\  & \underline{1} \\  & 2 \\ \end{align} \,}} \right.  \\ \end{align}$$Step 3:Bring down the next pair. Here our next pair is 200.$$\begin{align}  & 1 \\  & 1\left| \!{\overline {\,  \begin{align}  & \overline{3}.\overline{00}\overline{00}\overline{00}\overline{00} \\  & \underline{1} \\  & 200 \\ \end{align} \,}} \right.  \\ \end{align}$$Step 4:Now, double the value of quotient and write it in divisor. For the second digit, we need to write such number which when multiplied by the new number obtained gives us less than or equal to dividend.Here the quotient is 1. So, one digit of the divisor will be 2. Now, if we take the third digit 7 and then multiply 27 by 7, we get the product equal to our dividend.$\Rightarrow 27\times 7=189$$$\begin{align}  & 1.7 \\  & 1\left| \!{\overline {\,  \begin{align}  & \overline{3}.\overline{00}\overline{00}\overline{00}\overline{00} \\  & \underline{1} \\  & 200 \\ \end{align} \,}} \right.  \\  & 27\left| \!{\overline {\,  \begin{align}  & 200 \\  & \underline{189} \\  & 11 \\ \end{align} \,}} \right.  \\ \end{align}$$We keep on doing the process till 6 decimal places.$$\begin{align}  & 1.73 \\  & 343\left| \!{\overline {\,  \begin{align}  & 1100 \\  & \underline{1029} \\  & 71 \\ \end{align} \,}} \right.  \\ \end{align}$$$$\begin{align}  & 1.732 \\  & 3462\left| \!{\overline {\,  \begin{align}  & 7100 \\  & \underline{6924} \\  & 176 \\ \end{align} \,}} \right.  \\ \end{align}$$$$\begin{align}  & 1.7320 \\  & 34640\left| \!{\overline {\,  \begin{align}  & 17600 \\  & \underline{0} \\  & 17600 \\ \end{align} \,}} \right.  \\ \end{align}$$$$\begin{align}  & 1.73205 \\  & 346405\left| \!{\overline {\,  \begin{align}  & 1760000 \\  & \underline{1732025} \\  & 27975 \\ \end{align} \,}} \right.  \\ \end{align}$$$$\begin{align}  & 1.732050 \\  & 3464100\left| \!{\overline {\,  \begin{align}  & 2797500 \\  & \underline{0} \\  & 2797500 \\ \end{align} \,}} \right.  \\ \end{align}$$Hence, the square root of 3 is $$1.732050$$.So, the correct answer is “$$1.732050$$”.Note: The long-division method and arranging the set of 6 digits is different for integer and decimal. But taking double for the next division and putting a particular number is the same process for both of them. Since 3 is a non-perfect square number, we will find the value of root 3 using the long division method as shown above.