Exercise 6.3 — Division Method

Exercise 6.3 – Square Root by Long Division Method

Exercise 6.3 of Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) introduces the long division method for finding square roots — a powerful technique that works for any number, including large numbers and decimals, where prime factorisation becomes slow or impossible. This exercise has 8 questions covering four types of problems: finding square roots of integers, finding square roots of decimals, adjusting a number by subtraction or addition to make it a perfect square, and estimating square roots to the nearest whole number.

Unlike the prime factorisation method from Exercise 6.2, the long division method works even when a number has large or awkward prime factors. It is a key technique tested in board exams across Telangana, Andhra Pradesh, and CBSE Class 8.

How the Long Division Method Works — Step-by-Step

The long division method for square roots is built on a simple idea: group the digits in pairs starting from the decimal point (or from the right for whole numbers), then find how many times a carefully chosen divisor fits into each successive group. Here is the complete procedure:

Question 1 — Find Square Roots by Long Division Method

Question 2 — Square Roots of Decimal Numbers

For decimal numbers, pair the digits on each side of the decimal point separately — going left for the integer part and going right for the decimal part. Place the decimal point in the square root exactly when you bring down the first decimal pair. The method is otherwise identical to the integer case.

Questions 3 to 7 — Applying the Division Method

Question 8 — Estimate Square Roots to the Nearest Whole Number

When a number is not a perfect square, you can estimate its square root by identifying which two consecutive perfect squares it falls between, then choosing the nearer one. This is a quick mental technique useful in CBSE and state board MCQs and estimation questions.

Method: Find n such that n² ≤ given number < (n+1)². If the number is closer to n², √ ≈ n; if closer to (n+1)², √ ≈ n+1.

Exercise 6.3 — Quick Answer Summary

Common Mistakes to Avoid in Exercise 6.3

• Wrong digit grouping: Always pair digits from the decimal point outward. "1089" groups as "10|89", not "1|08|9". One wrong group ruins the entire calculation.
• Forgetting to double the quotient for the trial divisor: The trial divisor is always 2 × (current quotient), not the quotient itself. Missing this step is the most common error in board exams.
• Decimal square roots — wrong placement of decimal point: The decimal point goes in the answer exactly when you cross the decimal point in the number. Practice this with 2(i)–(iv) until it is automatic.
• Subtract vs Add questions — confusing the two: For Q3 (subtract), the answer is the remainder in the division. For Q7 (add), the answer is next perfect square minus the given number. These are opposite operations.
• Estimation — choosing the wrong square: Always compute both differences (number − lower square) and (upper square − number) and pick the smaller one. Do not guess by eye.

What Exercise 6.3 Prepares You For

The long division method is the last major technique in Chapter 6. Together with the prime factorisation method from Exercise 6.2, it gives you a complete toolkit for handling square roots at Class 8 level.

The estimation skill from Question 8 directly connects to irrational numbers in Class 9, where you locate surds like √2 and √3 on the number line by identifying which consecutive integers they fall between — the exact same reasoning used here. The subtraction and addition technique (Questions 3 and 7) also reappears in completing the square for quadratic equations in Class 9 and 10.