How to find the square root by division method?
Understand the Problem
The question is asking for a method to calculate the square root of a number using the division method. This involves a stepbystep process that includes pairing digits, estimating values, and performing division.
Answer
The square root is approximately $15.7$.
Answer for screen readers
The square root of the number calculated step by step is approximately $15.7$.
Steps to Solve

Pair the digits from the right Write the number for which you want to find the square root. Starting from the decimal point (or the rightmost digit if there is no decimal), pair the digits in groups of two. For example, for the number 152.25, it would be paired as (1)(52)(.)(25).

Determine the largest integer for the first pair Look at the leftmost pair (or single digit if there are an odd number of digits). Determine the largest integer whose square is less than or equal to this number. For 1, since $1^2 = 1$, the largest integer is 1.

Subtract and bring down the next pair Subtract the square of the integer you found from the pair, then bring down the next paired digits. For example: $$ 1  1 = 0 $$ Now, bring down the next pair, which is 52, making it 052.

Double the current quotient and find a new digit Double the number you have in the quotient (which is 1), and write it down with a blank on its right. This gives you 2_. You have to find a digit (let's call it x) such that $2x \cdot x$ is less than or equal to 52.

Estimate the new digit By testing values, you can find that $2 \cdot 6 \cdot 6 = 72$ (too high), but $2 \cdot 5 \cdot 5 = 50$ (just right). So x = 5.

Repeat the process Now, you have 15 as a quotient. Subtract 50 from 52: $$ 52  50 = 2 $$ Bring down the next pair (25), making it 225.

Continue with more digits Now, double the quotient (15) to get 30_. Find a digit (let's call it y) such that $30y \cdot y$ is less than or equal to 225. Testing 7 gives $30 \cdot 7 \cdot 7 = 147$; testing 8 gives $30 \cdot 8 \cdot 8 = 192$; but testing 9 gives $30 \cdot 9 \cdot 9 = 243$ (too high). So y = 7.

Calculate the result Now update the quotient to 15.7. Subtract 147 from 225: $$ 225  147 = 78 $$ You can stop here or continue with more pairs to get a more precise answer.
The square root of the number calculated step by step is approximately $15.7$.
More Information
The division method for finding square roots is an ancient technique that provides a way to manually compute square roots, which helps in understanding the relationship between numbers.
Tips
 Forgetting to properly pair the digits, which can lead to incorrect calculations.
 Miscalculating the square of the guessed digit, leading to errors in the subtraction step.
 Not updating the double of the quotient correctly after each iteration.