How to Find the Square Root of 35 by Long Division Method with Step-by-Step Instructions. Please answer, no spam please
The Correct Answer and Explanation is:
To find the square root of 35 using the long division method, follow the detailed steps below. This method helps us get a precise decimal value of the square root. The square root of 35 is not a perfect square, so we will get a non-terminating decimal value.
Step-by-step Long Division Method
- Group the digits in pairs starting from the decimal point. For 35, we write it as 35.00 00 00 to get accuracy up to 3 decimal places.
- Find the largest number whose square is less than or equal to 35.
- 6 × 6 = 36 (too big)
- 5 × 5 = 25 (works)
So, the first digit is 5.
Write 5 as the quotient and 25 below 35. Subtract: 35 – 25 = 10.
- Bring down two zeros to make it 1000.
Now, double the quotient (5) → 5 × 2 = 10. Write this as the new divisor prefix: 10_. - Find a digit X such that (10X × X) ≤ 1000.
Try:- 101 × 1 = 101
- 102 × 2 = 204
- 103 × 3 = 309
- 104 × 4 = 416
- 105 × 5 = 525
- 106 × 6 = 636
- 107 × 7 = 749
- 108 × 8 = 864
- 109 × 9 = 981 (fits)
So, X = 9. Add 9 to the quotient: now it’s 5.9. Subtract 981 from 1000 → remainder = 19.
- Bring down two more zeros → 1900.
Double the current quotient (59) → 59 × 2 = 118. New divisor prefix: 118_. - Find a digit Y such that (118Y × Y) ≤ 1900.
Try:- 1181 × 1 = 1181
- 1182 × 2 = 2364 (too big)
So, Y = 1. Add 1 to the quotient → now it’s 5.91. Subtract 1181 from 1900 → remainder = 719.
Conclusion and Explanation
The long division method is a classical approach to finding square roots, especially useful when the number is not a perfect square. In this case, we applied it to the number 35. The method starts by pairing digits and identifying the highest square less than the number. This gives us the initial digit of the root.
We began with 5 because 5² = 25, which is closest to 35 without going over. After subtracting, we brought down pairs of zeros and used a modified divisor formed by doubling the current quotient. This divisor prefix is then extended by trial digits until the product remains less than or equal to the dividend segment.
The process is repeated to calculate more decimal places, each time increasing the precision of the square root. Through the method, we found the square root of 35 ≈ 5.91 to two decimal places, and it can be extended further.
This method is particularly helpful in manual calculations without a calculator. It builds an understanding of how square roots behave, especially with irrational numbers like √35 which cannot be simplified into whole numbers or exact fractions. The accuracy depends on how many steps you continue the process.
Thus, √35 ≈ 5.916 if extended to three decimal places. The long division method, though a bit time-consuming, is reliable and provides a clear logical progression in finding square roots manually.
