Question

What is the square root of 164?

Answer 1

Hint: First we will write the given number as the product of its prime factors. Now we will try to form pairs of two similar prime factors and write then in exponential form with exponent equal to 2. The factors which may not be paired will be left inside the under root sign whose value we will find using the long division method up to three decimal places.

Complete step by step solution:
Here we have been asked to find the square root of 164.
First let us check if we can use the prime factorization method to get the root. So writing 164 as the product of its prime factors we get,
$\begin{array}{rl}& \Rightarrow 164=2\times 2\times 41\\ & \Rightarrow 164=2^2\times 41\end{array}$
Taking square root both the sides we get,
$\begin{array}{rl}& \Rightarrow \sqrt{164}=\sqrt{2^2\times 41}\\ & \Rightarrow \sqrt{164}=\sqrt{2^2}\times \sqrt{41}\end{array}$
Here we can take the factor 2 out of the root by using the formula ${\left(a^m\right)}^n=a^{m\times n}$ so we get,
$\Rightarrow \sqrt{164}=2\sqrt{41}$
As 41 cannot be factored further because it is a prime number so we need to apply the long division method to find the square root of 41. So let us see each step while finding the root.
Step (1): Starting from the rightmost digit we will form pairs of two digits till all the digits are paired or a single digit is left at the leftmost place, here we have only two digits in 41 so they will get paired.
Step (2): We will select a number whose square will be less than or equal to 41. Here we have to select 6. Now, we will write 6 as the divisor and the quotient and subtract the product from 41. The remainder will be 5.
$$\begin{array}{rl}& 6\\ & \begin{array}{c}6\\ +6\end{array}|\overline{\begin{array}{rl}& 41\\ & -36\\ & \stackrel{―}{5}\end{array}}\end{array}$$
Step (3): Now, the new divisor will be the sum 6 + 6 = 12. Clearly we can see that we cannot divide 5 by 12 so we will take a decimal point after 6 is the quotient and put two 0’s after 5 in the remainder.
$$\begin{array}{rl}& 6.\\ & \begin{array}{c}6\\ +6\\ \overline{12}\end{array}|\overline{\begin{array}{rl}& 41\\ & -36\\ & \stackrel{―}{500}\end{array}}\end{array}$$
Step (4): Now, we need to select a digit after 12 such that when we will take the product of this selected digit with the new divisor we will get a number less than 500. So we have to select 4. Now the new divisor will become 124 + 4 = 128. The quotient will become 6.4 and the remainder will become 500 – 496 = 4.
$$\begin{array}{rl}& 6.4\\ & \begin{array}{c}6\\ +6\\ \overline{124}\\ +4\\ \overline{128}\end{array}|\overline{\begin{array}{rl}& 41\\ & -36\\ & \stackrel{―}{500}\\ & -496\\ & \stackrel{―}{4}\end{array}}\end{array}$$
Step (5): In this step we will take two 0’s after 4 to get 400. This is due to the decimal point because we cannot divide 4 by 128. Now, we need to select a digit after 128 such that when it is multiplied with the same selected digit it gives a number less than 400. Clearly we can see that there will not be any such positive integer so we will take 0 after 128 and 6.4 and due to this we have to take two 0’s after 400. The new quotient will become 6.40 and the new divisor will become 1280 + 0 = 1280.
$$\begin{array}{rl}& 6.4\\ & \begin{array}{c}6\\ +6\\ \overline{124}\\ +4\\ \overline{128}\end{array}|\overline{\begin{array}{rl}& 41\\ & -36\\ & \stackrel{―}{500}\\ & -496\\ & \stackrel{―}{400}\end{array}}\end{array}$$
$$\begin{array}{rl}& 6.40\\ & \begin{array}{c}6\\ +6\\ \overline{124}\\ +4\\ \overline{1280}\\ +0\\ \overline{1280}\end{array}|\overline{\begin{array}{rl}& 41\\ & -36\\ & \stackrel{―}{500}\\ & -496\\ & \stackrel{―}{40000}\\ & -0\\ & \stackrel{―}{40000}\end{array}}\end{array}$$
Step (6): Again we will perform a similar process like step 2 and select a number after 1280 such that its product with the selected digit should be less than 40000. So, we have to select 3 and therefore the new quotient will be 6.403 which will be our answer.
$$\begin{array}{rl}& 6.403\\ & \begin{array}{c}6\\ +6\\ \overline{124}\\ +4\\ \overline{1280}\\ +0\\ \overline{12803}\\ +3\\ \overline{12806}\end{array}|\overline{\begin{array}{rl}& 41\\ & -36\\ & \stackrel{―}{500}\\ & -496\\ & \stackrel{―}{40000}\\ & -0\\ & \stackrel{―}{40000}\\ & -38409\\ & \stackrel{―}{1591}\end{array}}\end{array}$$
So the square root of 164 will be given as:
$\begin{array}{rl}& \Rightarrow \sqrt{164}=2\times 6.403\\ & \therefore \Rightarrow \sqrt{164}=12.806\end{array}$

Hence, the square root of 164 is 12.806 correct up to three decimal places of decimal.

Note: We can also find the square root of 164 directly using the long division method but we have used the prime factorization at the initial step. The only reason is that after the prime factorization we have only two digits left inside the radical sign which makes our calculation easier. In case you want to use the long division method for 164 then all the steps will be similar but at the initial step we will pair 4 and 6 while leaving 1 alone.