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REET Level 1+2 . Maths

Squares and Square Roots - REET Maths Notes

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Squares and Square Roots spans everything from spotting perfect square numbers to finding the square root of a surd in REET Maths. This page brings together all the key properties, methods, and REET 2024's solved questions in one place.

  • 2 questions in REET 2024
  • Class 8 level
  • 6 min read
Verified 17 September 2026 · Source: NCERT Class 6-8 Maths textbooks (official)

How REET asks this topic

REET 2024 had two very different questions on this topic - one required finding the largest perfect square number with a given number of digits (5 digits), the other required finding the square root of a number given as a surd (7+4*sqrt(3)). In other words, memorising the squares table isn't enough here - both estimation skills and the use of algebraic identities are tested.

Squares from 1 to 20

NumberSquareNumberSquare
1111121
2412144
3913169
41614196
52515225
63616256
74917289
86418324
98119361
1010020400

For calculation speed in REET, the squares from 1-20 should be committed to memory.

Properties of perfect square numbers

A perfect square number never ends in 2, 3, 7 or 8 - this is the fastest screening trick in multiple-choice questions. A number can only be a perfect square if it ends in an even number of zeros (like 100, 10000).

The sum of the first n odd natural numbers is always equal to n^2 - for example, the sum of the first 5 odd numbers (1+3+5+7+9) is 25 = 5^2. The square of an even number is even, and the square of an odd number is odd.

Per REET 2024's question, the largest 5-digit number is 99999, and sqrt(99999) is approximately 316.2, so 316^2 = 99856 is the largest 5-digit perfect square, because 317^2 = 100489 is already a 6-digit number.

Methods for finding a square root

In the prime factorisation method, the number is split into pairs of prime factors - for example, 324 = 2x2x3x3x3x3 = 2^2 x 3^4, so sqrt(324) = 2x9 = 18. This method only works for perfect square numbers.

The long division method is used to find the square root of any number (including decimals), and in REET it's the main method for finding square roots of large numbers.

A square root given as a surd (like 7+4*sqrt(3)) is solved by writing it in the form (a+b)^2 = a^2+2ab+b^2. Here, (2+sqrt(3))^2 = 4 + 4*sqrt(3) + 3 = 7+4*sqrt(3), so the square root of 7+4*sqrt(3) is 2+sqrt(3).

An example of the long division method: finding sqrt(841) gives an answer of 29, because 29x29=841. This method finds the square root of any number (large or decimal) step by step without needing factorisation, so it's useful for general calculation practice in REET too.

Pythagorean triplets

Formula
For any natural number m > 1, the numbers 2m, m^2-1, m^2+1 form a Pythagorean triplet.
Example (m=2)
2m=4, m^2-1=3, m^2+1=5, giving the triplet (3, 4, 5); check: 3^2+4^2=9+16=25=5^2.
Example (m=3)
2m=6, m^2-1=8, m^2+1=10, giving the triplet (6, 8, 10); check: 6^2+8^2=36+64=100=10^2.
Example (m=4)
2m=8, m^2-1=15, m^2+1=17, giving the triplet (8, 15, 17); check: 8^2+15^2=64+225=289=17^2.

Estimating the number of digits in a square root

You can estimate how many digits a square root will have from how many digits the perfect square number has: if the perfect square number has n digits and n is even, the square root will have n/2 digits; if n is odd, the square root will have (n+1)/2 digits.

For example, the square root of a 9-digit (odd) number will have (9+1)/2 = 5 digits, while the square root of a 4-digit (even) number will have 4/2 = 2 digits. This rule helps you quickly eliminate wrong answer options for large numbers' square roots without doing the full long division.

Approximate values of common irrational square roots

sqrt(2)
approximately 1.414
sqrt(3)
approximately 1.732
sqrt(5)
approximately 2.236
sqrt(7)
approximately 2.646

Why memorise these values

The approximate decimal values of irrational square roots like sqrt(2), sqrt(3), sqrt(5) come in handy for quick estimation in geometry (like finding a diagonal's length) and questions that simplify surds. Once memorised properly, they let you quickly sanity-check answer options in multiple-choice questions, especially when an exact solution isn't necessary.

How to teach this to children

Classroom activity

In Class 8, have students build a square arrangement of n x n dots on dot-grid or graph paper - counting 3x3, 4x4 and so on lets children see for themselves why a square number is called 'square', making the link between number and shape concrete.

Teaching aids

Put up a chart of squares from 1-20 and coloured square shapes drawn on graph paper in class, so children remember square numbers through repeated visual exposure.

Common misconception

Children often confuse 'square' (n^2) with 'double' (2n), or think every number's square root is a whole number. Clarify both misconceptions repeatedly with examples.

Assessment idea

Have students practise finding the nearest perfect square above and below a given number (like 200) - this builds estimation skills, which apply directly in REET-style questions.

Cross-curricular link

Use practical examples of the Pythagorean theorem (like the distance between a ladder and a wall) to connect squares and square roots to geometry, so children see their real-world use.

Trick to remember / exam tip

The units digit of a perfect square number immediately tells you the possible units digit of its square root: a units digit of 1 gives a square root ending in 1 or 9, a units digit of 4 gives 2 or 8, a units digit of 9 gives 3 or 7, a units digit of 6 gives 4 or 6, and a units digit of 5 gives 5 - this lets you instantly rule out wrong options.

Questions asked in REET 2024 (with solutions)

Question 115

The largest five-digit perfect square number is :

  1. A10000
  2. B99586
  3. C99999
  4. D99856

Explanation

पाँच अंकों की सबसे बड़ी संख्या 99999 है और √99999 ≈ 316.2। अतः 316² = 99856 पाँच अंकों की सबसे बड़ी पूर्ण वर्ग संख्या है (317² = 100489 छह अंकों की है)। 10000 सबसे छोटी पाँच अंकीय पूर्ण वर्ग संख्या है।

Question 115

Square root of 7 + 4√3 is

  1. A2 + √3
  2. B2 − √3
  3. C4 + 2√3
  4. D61 + 28√3

Explanation

(2 + √3)² = 2² + 2×2×√3 + (√3)² = 4 + 4√3 + 3 = 7 + 4√3। अतः 7 + 4√3 का वर्गमूल 2 + √3 है। (2 − √3)² = 7 − 4√3 होता है और (D) तो 7 + 4√3 का वर्ग है।

Practice Questions

Question 1

What is the value of 15²?

  1. A205
  2. B215
  3. C225
  4. D235

Explanation

15x15=225.

Question 2

Which of the following is not a perfect square?

  1. A144
  2. B169
  3. C186
  4. D196

Explanation

144=12^2, 169=13^2, 196=14^2 are all perfect squares, but 186 is not the square of any integer (13^2=169, 14^2=196, and 186 falls in between).

Question 3

What is the value of √225?

  1. A12
  2. B13
  3. C15
  4. D25

Explanation

15x15=225, so sqrt(225)=15.

Question 4

The unit digit of a perfect square can never be:

  1. A1
  2. B4
  3. C8
  4. D9

Explanation

Perfect square numbers can only end in 0,1,4,5,6,9; never 2,3,7,8.

Question 5

What is the value of √(11 + 6√2)?

  1. A3 + √2
  2. B3 − √2
  3. C2 + 3√2
  4. D9 + 2√2

Explanation

(3+sqrt(2))^2 = 9 + 6*sqrt(2) + 2 = 11 + 6*sqrt(2), so the square root of 11+6*sqrt(2) is 3+sqrt(2).

Question 6

What is the sum of the first 10 odd natural numbers?

  1. A90
  2. B100
  3. C110
  4. D55

Explanation

The sum of the first n odd numbers is n^2, so for n=10, sum = 10^2=100.

Question 7

Which of the following is not a Pythagorean triplet?

  1. A(3, 4, 5)
  2. B(6, 8, 10)
  3. C(5, 12, 13)
  4. D(4, 5, 7)

Explanation

4^2+5^2=16+25=41, while 7^2=49 - the two are not equal, so (4,5,7) is not a Pythagorean triplet. The other three triplets all satisfy a^2+b^2=c^2.

Question 8

What is the smallest 4-digit perfect square number?

  1. A1000
  2. B1024
  3. C961
  4. D1089

Explanation

31^2=961 (3-digit), 32^2=1024 (the smallest 4-digit perfect square).

Question 9

The area of a square garden is 2116 sq. m. What is the length of one side of the garden?

  1. A44 मीटर
  2. B46 मीटर
  3. C48 मीटर
  4. D52 मीटर

Explanation

Side = sqrt(2116) = 46 metres, because 46x46=2116.

Question 10

Which is the most suitable activity to concretely explain the concept of square numbers to Class 8 students?

  1. Aडॉट/ग्राफ पेपर पर n×n बिंदुओं की वर्गाकार सज्जा बनवाना
  2. Bकेवल सूत्र याद कराना
  3. Cपाठ्यपुस्तक के उदाहरण पढ़वाना
  4. Dकक्षा में केवल परीक्षा देना

Explanation

Visual, concrete material (dot arrays) helps children understand the geometric meaning of a square number, which gives far more effective and lasting learning than rote memorisation.

Frequently Asked Questions

How many questions come from Squares & Square Roots in REET?+

REET 2024 had 1-2 questions from this topic in the Maths section, based on digit-count limits and surd square roots.

Which digit can a perfect square number never end in?+

A perfect square number can never end in 2, 3, 7 or 8.

What are the main methods for finding a square root?+

The prime factorisation method and the long division method - both can be asked at REET level.

What is a Pythagorean triplet?+

Three positive integers (a, b, c) that satisfy a^2+b^2=c^2 are called a Pythagorean triplet - like (3,4,5) or (6,8,10).

What is the approximate value of sqrt(2)?+

The approximate value of sqrt(2) is 1.414 - it's an irrational number, whose decimal expansion never terminates or repeats.

Does every number have a whole-number square root?+

No, only perfect square numbers (like 4, 9, 16) have whole-number square roots; other numbers have irrational square roots (like sqrt(2), sqrt(3)).

Does a negative number have a real square root?+

No, the square root of a negative number is not defined within the real numbers, because no real number's square can be negative.

Are squaring and square-rooting inverse operations of each other?+

Yes, if a^2=b, then the positive root gives sqrt(b)=a - for example, 12^2=144, so sqrt(144)=12. The two operations cancel each other out.

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Indices

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Cubes and Cube Roots

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REET Syllabus

See the complete REET Level 1 and Level 2 syllabus in one place.

REET 2024 Solved Paper

Read all REET 2024 questions with solutions and explanations.

Official Sources

  1. NCERT Class 6-8 Maths textbooks (official)
  2. Rajasthan Board of Secondary Education (BSER) - official REET information

Change Log

  • 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked