MathsICSE Class 8 · Squares and Square Roots

Square Root by the Long Division Method: Decimals and Nearest Squares (ICSE Class 8)

Q: Why double the quotient?

Because (10t + d)^2 = 100t^2 + 20td + d^2; the 20t part is the doubled quotient.

Q: Can the method find roots of non-squares?

Yes, to any decimal place, by bringing down pairs of zeros.

Updated 17 October 2026 · Fact-checked

Pair the digits in twos from the decimal point outwards, find the largest square at most the first pair, subtract, bring down the next pair, double the quotient so far, and choose the next digit so that (double with the digit) multiplied by the digit fits. Repeat. For a non-square, the remainder tells you the least number to subtract.

Understand Square Root by Long Division

When a number is large, or is a decimal, factor trees are slow. The long division method works digit by digit.

Pairing. Starting from the decimal point, group digits in pairs: to the left for the integer part, to the right for the decimal part. For $7396$ the pairs are $73 ∣ 96$ ; for $51.84$ they are $51.84$ . If the last pair on the right has one digit, add a $0$ .

Steps.
1. Find the largest square at most the first pair; its root is the first digit. Subtract.
2. Bring down the next pair next to the remainder.
3. Double the quotient so far and write it as a trial divisor with a blank digit. Choose the largest digit $d$ such that (trial divisor with $d$ ) $\times d$ does not exceed the dividend.
4. Subtract and repeat. Put the decimal point in the root when you bring down the first decimal pair.

For a decimal with two places, the root has one decimal place; with four places, two.

Least number to subtract. If the remainder at the end is $r \neq = 0$ , then $N - r$ is a perfect square, so $r$ is the least number to subtract. Least number to add. If $N$ lies between $k$ and $k + 1$ , then $(k + 1)^{2} - N$ is the least number to add.

Estimating. Roots are bounded by neighbouring squares: $7 8^{2} = 6084$ and $7 9^{2} = 6241$ , so $6200$ is between $78$ and $79$ . The CISCE limit is at most four digits in total and at most two decimal places.

Square Root by Long Division: formulas to remember

Pairing rule: $pairs from the decimal point outward$; Integer part leftwards, decimal part rightwards.
Least number to subtract: $N - k^{2}, k = ⌊ N ⌋$; This is the remainder of the long division.
Least number to add: $(k + 1)^{2} - N$; For a non-square $N$ .

How to solve Square Root by Long Division questions

Pair, divide, double, repeat; use neighbouring squares for add/subtract questions.

1Pair the digits from the decimal point.
2Take the largest square at most the first pair; subtract.
3Bring down the next pair; double the root so far for the trial divisor.
4Pick the next digit; subtract; repeat until pairs are used up.
5Check by squaring; for non-squares, read the remainder or use the next square.

Square Root by Long Division in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Square field

Suppose a square garden has area 92.16 square metres and fencing costs Rs 25 per metre.

Find the cost of fencing the garden.

Try it first, then show the solutionSolution

Side $= 92.16 = 9.6$ m (since $9. 6^{2} = 92.16$ ).
Perimeter $= 4 \times 9.6 = 38.4$ m; cost $= 38.4 \times 25 = 960$ .

Answer: Rs 960.

Memory hook: Area gives the side by a square root; perimeter is four sides.

Tricks to remember Square Root by Long Division

Short memory aids. Say each one aloud once, then use it on a question the same day.

Two digits, one digit
Each pair of digits in the number gives one digit in the root.
Use it when: Counting digits of the answer.
Neighbour squares
Remember squares of 70 to 99 near the target to see the add/subtract answer instantly.
Use it when: Least number problems.

Square Root by Long Division: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Decimal root, MCQ · 1 mark · MCQBoard level

$6.25$ is:
(a) 0.25 (b) 2.5 (c) 25 (d) 0.625

Try it first, then show the solutionSolution

Pairs $6.25$ ; $2^{2} = 4$ , remainder $2$ ; bring $25$ : $225$ ; trial $4_\times_$ : $45 \times 5 = 225$ .
Root $= 2.5$ .

Answer: (b)

Example 2: Four-digit root · 3 marks · Short answerBoard level

Find $7396$ by the long division method.

Try it first, then show the solutionSolution

Pairs: $73 ∣ 96$ . Largest square $\leq 73$ is $64$ , root digit $8$ , remainder $9$ .
Bring down $96$ : $996$ . Double $8$ gives $16_$ ; $166 \times 6 = 996$ .
Remainder $0$ , root $= 86$ . Check: $8 6^{2} = 7396$ .

Answer: 86

Example 3: Non-square, two decimal places · 3 marks · Short answerHarder board level

Find $7$ correct to two decimal places by long division.

Try it first, then show the solutionSolution

Write $7.00 0000$ in pairs. $2^{2} = 4$ ; remainder $3$ .
Bring $00$ : $300$ . Double $2$ is $4_$ ; $46 \times 6 = 276$ ; remainder $24$ ; digit $6$ .
Bring $00$ : $2400$ . Double $26$ is $52_$ ; $524 \times 4 = 2096$ ; remainder $304$ ; digit $4$ .
Bring $00$ : $30400$ . Double $264$ is $528_$ ; $5285 \times 5 = 26425$ ; digit $5$ . So $7 = 2.645 \dots$ , which rounds to $2.65$ .
Check: $2.6 4^{2} = 6.9696$ , $2.6 5^{2} = 7.0225$ ; $7$ is nearer to $2.6 5^{2}$ .

Answer: 2.65

Example 4: Least to add and subtract · 4 marks · Short answerHarder board level

Find (i) the least number to be added to 6200 and (ii) the least number to be subtracted from 6200 to get a perfect square.

Try it first, then show the solutionSolution

$7 8^{2} = 6084$ and $7 9^{2} = 6241$ ; so $78 < 6200 < 79$ .
(i) Add $6241 - 6200 = 41$ (new root $79$ ).
(ii) Subtract $6200 - 6084 = 116$ (new root $78$ ).

Answer: (i) 41; (ii) 116

Example 5: Smallest four-digit square · 4 marks · Long answerHOTS / competency

Find the smallest 4-digit perfect square, its square root, and the least number to be added to 1000 to reach it.

Try it first, then show the solutionSolution

The smallest 4-digit number is $1000$ . Pairs $10 ∣ 00$ : $3^{2} = 9$ , remainder $1$ ; $100$ ; $61 \times 1 = 61$ , so $1000 \approx 31.6$ .
So $3 1^{2} = 961 < 1000$ and $3 2^{2} = 1024$ ( $3 0^{2} + 2 \cdot 30 \cdot 2 + 2^{2} = 900 + 120 + 4$ ).
Smallest 4-digit square $= 1024$ , root $32$ ; number to add $= 1024 - 1000 = 24$ .

Answer: 1024; root 32; add 24

Where students lose marks in Square Root by Long Division

The slips examiners see most often, and what to write instead.

Mistake: Pairing digits from the left in a decimal
Pairs must start at the decimal point.
Wrong working
$15.9201 \to 1 ∣ 59 ∣ 20 ∣ 1$ .
Right working
$15.92 ∣ 01$ .
Do this: Mark the decimal point first, then pair outward.
Usually costs: The entire root.
Mistake: Forgetting to double the quotient
The new divisor is twice the root so far, not the root itself.
Wrong working
After root digit $8$ , trial divisor $8_$ .
Right working
Trial divisor $16_$ .
Do this: Write "double" next to every divisor line.
Usually costs: Method marks.

Exam tips

Show every subtraction line; partial credit depends on it.
Check the root by squaring or by estimating between two neighbouring squares.
Position the decimal point in the root as soon as you bring down the first decimal pair.

Practice questions on Square Root by Long Division

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
$0.0049$ equals:
(a) 0.7 (b) 0.07 (c) 0.007 (d) 0.0007
Hint
Four decimal places give a root with two.
Answer
(b)
Q2 · 3 marks · Short answerBoard level
Find $2916$ by long division.
Hint
Pairs 29|16.
Answer
$5^{2} = 25$ , remainder $4$ ; bring $16$ : $416$ ; $104 \times 4 = 416$ . Root $= 54$ . Check $5 4^{2} = 2916$ .
Q3 · 3 marks · Short answerHarder board level
The area of a square plot is 92.16 m $^{2}$ . Find its side and the cost of fencing it at Rs 40 per metre.
Hint
Perimeter is 4 times the side.
Answer
Side $= 9.6$ m (since $9. 6^{2} = 92.16$ ). Perimeter $= 38.4$ m; cost $= 38.4 \times 40 = 1536$ . Rs 1536.
Q4 · 3 marks · Short answerHarder board level
Find the least number to be added to 7000 so that the sum is a perfect square, and the square root of the sum.
Hint
$8 3^{2} = 6889$ .
Answer
$8 4^{2} = 7056$ ; add $7056 - 7000 = 56$ ; root $= 84$ .
Q5 · 4 marks · Case-basedHOTS / competency
A school has 5000 chairs and arranges them in a square, with as many rows as chairs in a row. How many chairs are left over, and how many more chairs are needed to make the next larger square?
Hint
Find the two squares on either side of 5000.
Answer
$7 0^{2} = 4900$ and $7 1^{2} = 5041$ . Left over $= 5000 - 4900 = 100$ chairs; more needed $= 5041 - 5000 = 41$ chairs.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.How are digits paired in 15.9201?Show answer
15 . 92 | 01.
2.What is the square root of 6.25?Show answer
2.5.
3.Least number to add to make N a square?Show answer
(k+1)^2 - N where k is the integer part of the root.
4.How many digits does the root of a 4-digit number have?Show answer
Two.

Square Root by Long Division: frequently asked questions

Why double the quotient?

Because (10t + d)^2 = 100t^2 + 20td + d^2; the 20t part is the doubled quotient.

Can the method find roots of non-squares?

Yes, to any decimal place, by bringing down pairs of zeros.