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📐 Chapter 1 · NCERT 2025-26

🔸 A Square
and A Cube

Perfect Squares · Square Roots · Cubes · Cube Roots · Patterns

²
³
📐 Introduction

When you multiply a number by itself, you get its square. When you multiply a number by itself three times, you get its cube. These simple ideas lead to some of the most beautiful patterns in mathematics!

In this chapter, we will explore perfect squares and perfect cubes, learn how to find square roots and cube roots, discover amazing number patterns, and see how squares connect to the famous Pythagorean theorem.

🔶 What is a Square?

The square of a number n is n × n, written as n². For example, 5² = 5 × 5 = 25. The number 25 is called a perfect square.

📦 What is a Cube?

The cube of a number n is n × n × n, written as n³. For example, 4³ = 4 × 4 × 4 = 64. The number 64 is called a perfect cube.

📏 Square Root

The square root of a number is the value that, when multiplied by itself, gives the original number. √25 = 5 because 5 × 5 = 25.

📋 Cube Root

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. ³√64 = 4 because 4 × 4 × 4 = 64.
📚 Chapter Overview
Topic Key Concepts
Perfect Squares Properties, identifying perfect squares, unit digit patterns
Patterns in Squares Sum of odd numbers, difference of consecutive squares, palindromic patterns
Square Roots Prime factorization method, long division method
Perfect Cubes Properties, identifying perfect cubes
Cube Roots Prime factorization method
Pythagorean Triples Generating triples from squares, applications
💡 Why Study This? Squares and cubes appear everywhere — calculating areas and volumes, understanding growth patterns, cryptography, and even in music! This chapter lays the foundation for algebra and geometry in higher classes.
📊 Squares & Cubes of First 30 Numbers

This is a quick-reference table. Memorising squares up to 30 and cubes up to 15 will help you solve problems faster!

nn²n³nn²n³nn²n³
111111211331214419261
2481214417282248410648
39271316921972352912167
416641419627442457613824
5251251522533752562515625
6362161625640962667617576
7493431728949132772919683
8645121832458322878421952
9817291936168592984124389
1010010002040080003090027000
💡 Tip: Notice that perfect cubes end in the same unit digit as the number itself! For example: 7 ends in 7, and 7³ = 343 also ends in 3... wait, that’s not right — actually, 2³ = 8, 3³ = 27, 7³ = 343, 8³ = 512. The unit digits of cubes cycle through: 1, 8, 7, 4, 5, 6, 3, 2, 9, 0.
🔶 Perfect Squares & Their Properties

A natural number is called a perfect square if it can be expressed as the product of some natural number with itself. In other words, n is a perfect square if there exists a natural number m such that n = m².

📊 First 20 Perfect Squares
n12345678910
n²149162536496481100
n11121314151617181920
n²121144169196225256289324361400
🔎 Visual: Square as a Dot Array

A perfect square can be represented as a square arrangement of dots. Click the buttons below to see different square numbers as dot arrays.

Click to visualise a perfect square

3² = 9 dots arranged in a 3 × 3 grid

💡 Properties of Perfect Squares

② Unit Digit Pattern

A perfect square can only end in 0, 1, 4, 5, 6, or 9. It never ends in 2, 3, 7, or 8. So 247 and 163 can never be perfect squares!

③ Even & Odd Rule

The square of an even number is always even. The square of an odd number is always odd. Example: 6² = 36 (even), 7² = 49 (odd).

④ Zeros at the End

A perfect square always has an even number of trailing zeros (0, 2, 4, ...). So 900 (two zeros) is a perfect square, but 8000 (three zeros) is not.

⑤ Negative Numbers

The square of any real number is always non-negative. There is no real number whose square is negative. (−5)² = 25, not −25.

⑥ Between Consecutive Squares

Between n² and (n+1)², there are exactly 2n non-perfect-square numbers. Between 9 (3²) and 16 (4²), there are 2×3 = 6 numbers: 10, 11, 12, 13, 14, 15.

⑦ Digital Root

The digital root (repeated sum of digits) of a perfect square is always 1, 4, 7, or 9. This gives another quick check. Digital root of 144: 1+4+4 = 9 ✅
🔎 Number Grid: Spot the Perfect Squares (1–100)

Perfect squares are highlighted in amber. Can you see the pattern?

💡 Quick Test: To check if a number is a perfect square, first look at its unit digit. If it ends in 2, 3, 7, or 8, it is NOT a perfect square. If it ends in 0, 1, 4, 5, 6, or 9, it might be — you need to verify further using prime factorization.
🧠 Test Each Property — Quick Puzzles

For each property above, try these mini-challenges! Click Show Answer to check.

② Unit Digit Property — Puzzle

Q: Without calculating, which of these cannot be a perfect square?
A) 1764   B) 2023   C) 4096   D) 5184

Answer: B) 2023 — it ends in 3, and perfect squares never end in 2, 3, 7, or 8. (A = 42², C = 64², D = 72²)

③ Even/Odd Rule — Puzzle

Q: If a perfect square is odd, what can you say about the number that was squared?

Answer: The original number must be odd. Even² = Even, Odd² = Odd. So 49 is odd ⇒ √49 = 7 (odd).

④ Trailing Zeros — Puzzle

Q: Can 32000 be a perfect square? What about 3200?

Answer: 32000 has 3 trailing zeros (odd count) → NOT a perfect square. 3200 has 2 trailing zeros (even count) → might be, but 3200 / 100 = 32, and 32 is not a perfect square, so 3200 is also not a perfect square. The rule only eliminates; passing it doesn’t confirm.

⑤ Negative Numbers — Puzzle

Q: Is (−12)² = −144 or +144? Can −144 be a perfect square?

Answer: (−12)² = (−12) × (−12) = +144. Negative × Negative = Positive. So −144 is never a perfect square — no real number squared gives a negative result.

⑥ Between Consecutive Squares — Puzzle

Q: How many non-perfect-square numbers lie between 400 (= 20²) and 441 (= 21²)?

Answer: 2n = 2 × 20 = 40 numbers (401, 402, ... 440). Formula: between n² and (n+1)² there are 2n non-perfect-square numbers.

⑦ Digital Root — Puzzle

Q: Find the digital root of 2025. Could 2025 be a perfect square?

Answer: 2 + 0 + 2 + 5 = 9 → digital root = 9. Since 9 is in {1, 4, 7, 9}, it might be a perfect square. And indeed, 2025 = 45² ✅
🎲 Mixed Property Quiz

Use all the properties together. Pick the best reason to accept or reject each number.

💡 Amazing Patterns in Squares
🔶 Pattern 1: Sum of First n Odd Numbers

One of the most beautiful patterns in mathematics: the sum of the first n odd numbers always equals n²!

1 + 3 + 5 + 7 + ... + (2n − 1) = n² Sum of first n odd numbers = n squared
nOdd NumbersSumn²
1111 ✅
21 + 344 ✅
31 + 3 + 599 ✅
41 + 3 + 5 + 71616 ✅
51 + 3 + 5 + 7 + 92525 ✅
💡 Application: This means if you want to express 49 as a sum of odd numbers: 49 = 7², so it equals 1 + 3 + 5 + 7 + 9 + 11 + 13 (seven odd numbers). Conversely, if someone asks "Is 1 + 3 + 5 + ... + 19 a perfect square?" — yes! There are 10 odd numbers, so the sum is 10² = 100.

🎬 Watch: Odd Numbers Build a Square

✍ Practice — Pattern 1

Q1. Express 64 as a sum of consecutive odd numbers.

64 = 8², so it is the sum of the first 8 odd numbers:
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64

Q2. Find the value of 1 + 3 + 5 + 7 + ... + 31. Is it a perfect square?

The last odd number is 31 = 2n − 1, so n = 16.
Sum = 16² = 256. Yes, it is a perfect square!

Q3. How many odd numbers must be added to get 144?

144 = 12², so we need the first 12 odd numbers:
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 = 144
🔷 Pattern 2: Difference of Consecutive Squares

The difference between two consecutive perfect squares follows a simple rule:

(n + 1)² − n² = 2n + 1 Difference of consecutive squares is always an odd number
n(n+1)²n²Difference2n + 1
14133 ✅
29455 ✅
316977 ✅
4251699 ✅
536251111 ✅

🎬 Watch: Squares Grow by Odd Numbers

✍ Practice — Pattern 2

Q1. Find 17² − 16² without calculating each square.

Using (n+1)² − n² = 2n + 1, with n = 16:
17² − 16² = 2(16) + 1 = 33
Verify: 289 − 256 = 33 ✅

Q2. Find 100² − 99².

n = 99, so 100² − 99² = 2(99) + 1 = 199
Verify: 10000 − 9801 = 199 ✅

Q3. If the difference between two consecutive perfect squares is 41, what are the two squares?

2n + 1 = 41 ⇒ n = 20.
The two squares are 20² = 400 and 21² = 441.
441 − 400 = 41 ✅
🔸 Pattern 3: Sum of Consecutive Natural Numbers

The square of any odd number can be expressed as the sum of two consecutive natural numbers:

n² = (n² − 1)/2 + (n² + 1)/2   (when n is odd) Example: 7² = 49 = 24 + 25

3² = 9

9 = 4 + 5
(two consecutive numbers)

5² = 25

25 = 12 + 13
(two consecutive numbers)

9² = 81

81 = 40 + 41
(two consecutive numbers)

11² = 121

121 = 60 + 61
(two consecutive numbers)

✍ Practice — Pattern 3

Q1. Express 13² as a sum of two consecutive natural numbers.

13² = 169. Using the formula: (169 − 1)/2 = 84 and (169 + 1)/2 = 85.
169 = 84 + 85 ✅

Q2. Express 15² as a sum of two consecutive numbers.

15² = 225. (225 − 1)/2 = 112 and (225 + 1)/2 = 113.
225 = 112 + 113 ✅

Q3. The sum of two consecutive natural numbers is 169. What are the numbers? Which odd number’s square is this?

The two consecutive numbers are 84 and 85 (since 84 + 85 = 169).
169 = 13², so this is the square of the odd number 13.
🔹 Pattern 4: Palindromic Square Patterns

Look at these beautiful patterns involving 1s:

Repeating 1s Squared

1² = 1
11² = 121
111² = 12321
1111² = 1234321
11111² = 123454321
Notice: palindromes that count up and back down!

Patterns with 6

7² = 49
67² = 4489
667² = 444889
6667² = 44448889
More 6s in the number means more 4s and 8s in the square!
💡 Shortcut for Squaring Numbers Ending in 5:
To find (n5)² where n is any digit(s): multiply n by (n+1) and append 25.
Example: 35² = (3 × 4) followed by 25 = 1225
Example: 75² = (7 × 8) followed by 25 = 5625
Example: 125² = (12 × 13) followed by 25 = 15625

✍ Practice — Pattern 4

Q1. What is 111111²? (Six 1s squared.) Write the answer using the palindromic pattern.

Following the pattern (digits go up to the count of 1s, then back down):
111111² = 12345654321
1→2→3→4→5→6→5→4→3→2→1

Q2. Use the shortcut to find 45².

For 45²: n = 4, so n × (n+1) = 4 × 5 = 20, then append 25.
45² = 2025

Q3. Use the shortcut to find 95² and 105².

95²: n = 9, so 9 × 10 = 90 → append 25 → 9025
105²: n = 10, so 10 × 11 = 110 → append 25 → 11025

Q4. What is 66667²? Use the pattern with 6s.

Pattern: each extra 6 adds one more 4 and one more 8 in the square.
7² = 49, 67² = 4489, 667² = 444889, 6667² = 44448889
So 66667² = 4444488889 (five 4s, four 8s, ends in 9)
🔺 Pattern 5: Perfect Squares & Triangular Numbers

A triangular number is the number of dots that form an equilateral triangle. The nth triangular number T(n) = 1 + 2 + 3 + ... + n = n(n+1)/2.

n12345678910
T(n)13610152128364555

Here is the beautiful connection: the sum of two consecutive triangular numbers always gives a perfect square!

T(n) + T(n − 1) = n² Sum of consecutive triangular numbers = a perfect square
T(n−1)+T(n)=n²
T(1) = 1+T(2) = 3=4 = 2² ✅
T(2) = 3+T(3) = 6=9 = 3² ✅
T(3) = 6+T(4) = 10=16 = 4² ✅
T(4) = 10+T(5) = 15=25 = 5² ✅
T(5) = 15+T(6) = 21=36 = 6² ✅
💡 Why does this work?
T(n) = n(n+1)/2 and T(n−1) = (n−1)n/2.
Adding: n(n+1)/2 + (n−1)n/2 = n[(n+1) + (n−1)]/2 = n × 2n/2 = n²

🎬 Watch: Two Triangles Make a Square

🔬 Some Triangular Numbers are also Perfect Squares!

T(1) = 1 = 1²
T(8) = 36 = 6²
T(49) = 1225 = 35²
These are rare and special numbers!

✍ Practice — Pattern 5

Q1. Find T(7) + T(8). Is it a perfect square?

T(7) = 7×8/2 = 28, T(8) = 8×9/2 = 36.
T(7) + T(8) = 28 + 36 = 64 = 8² ✅
Yes! T(n−1) + T(n) always equals n².

Q2. What is the 12th triangular number? Verify that T(11) + T(12) = 12².

T(12) = 12×13/2 = 78
T(11) = 11×12/2 = 66
T(11) + T(12) = 66 + 78 = 144 = 12² ✅

Q3. Is T(15) a perfect square? Check using the formula.

T(15) = 15×16/2 = 120.
Is 120 a perfect square? 10² = 100, 11² = 121. Since 120 ≠ any n², T(15) is NOT a perfect square.
🔒 The Locker Puzzle

Imagine a school hallway with 100 lockers, all closed. 100 students walk past one by one:

Student 1

Toggles every locker
(opens all 100)

Student 2

Toggles every 2nd locker
(2, 4, 6, 8…100)

Student 3

Toggles every 3rd locker
(3, 6, 9, 12…99)

Student k

Toggles every k-th locker
(all the way to Student 100)

🤔 After all 100 students are done, which lockers are OPEN? Try it below!

Slow Fast
All 100 lockers are closed. Press "Next Student" to begin!
OPEN CLOSED Perfect Square
📏 Finding Square Roots

The square root of a number n is the value that, when multiplied by itself, gives n. We write it as √n. Finding the square root is the reverse of squaring.

① Prime Factorization Method

This is the most reliable method for finding square roots of perfect squares. The steps are:

  1. Find the prime factorization of the number.
  2. Group the prime factors into pairs.
  3. Take one factor from each pair.
  4. Multiply the chosen factors — this is the square root.

Example 1: Find √576 using prime factorization.

Step 1: Prime factorize 576.
576 = 2 × 288 = 2 × 2 × 144 = 2 × 2 × 2 × 72 = 2 × 2 × 2 × 2 × 36
= 2 × 2 × 2 × 2 × 2 × 18 = 2 × 2 × 2 × 2 × 2 × 2 × 9
= 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
576 = 26 × 3²
Step 2: Group into pairs.
576 = (2 × 2) × (2 × 2) × (2 × 2) × (3 × 3)
Step 3: Take one from each pair.
√576 = 2 × 2 × 2 × 3 = 24
Verification: 24 × 24 = 576 ✅

Example 2: Find √1764 using prime factorization.

Step 1: Prime factorize 1764.
1764 = 2 × 882 = 2 × 2 × 441 = 2 × 2 × 3 × 147
= 2 × 2 × 3 × 3 × 49 = 2 × 2 × 3 × 3 × 7 × 7
1764 = 2² × 3² × 7²
Step 2: Take one from each pair.
√1764 = 2 × 3 × 7 = 42
Verification: 42 × 42 = 1764 ✅
② Long Division Method

The long division method works for any number (even non-perfect squares) and is especially useful for large numbers. Here are the steps:

  1. Group the digits in pairs from right to left (the leftmost group may have 1 or 2 digits).
  2. Find the largest number whose square is ≤ the first group. This is the first digit of the answer.
  3. Subtract and bring down the next pair of digits.
  4. Double the quotient obtained so far, and find a digit d such that (doubled quotient followed by d) × d ≤ remainder with brought-down digits.
  5. Repeat until all pairs are used.

Example 3: Find √7056 by long division.

Step 1: Group digits in pairs from right.
7056 → 70 | 56 (two groups)
Step 2: First group = 70. Largest perfect square ≤ 70 is 64 (8²). So first digit = 8.
Subtract: 70 − 64 = 6. Bring down 56 to get 656.
Step 3: Double the quotient: 2 × 8 = 16_.
Find digit d such that 16d × d ≤ 656.
Try d = 4: 164 × 4 = 656. Perfect!
Answer: √7056 = 84
Verification: 84 × 84 = 7056 ✅

🎬 Long Division — Animated Step by Step

or
🔧 Interactive: Square Root Calculator
📏 Find the Square Root
Enter a number to see its square root calculated step by step using prime factorization.
Enter a number above and click "Find √" to see the step-by-step solution.
💡 When to Use Which Method:
• Prime Factorization: Best for perfect squares with small prime factors. Easy to understand and verify.
• Long Division: Works for any number, including non-perfect squares. Essential for finding approximate square roots.
📦 Perfect Cubes & Their Properties

A natural number is called a perfect cube if it can be expressed as the product of some natural number multiplied by itself three times. That is, n is a perfect cube if there exists a natural number m such that n = m³.

📊 First 15 Perfect Cubes
n12345678910
n³1827641252163435127291000
n1112131415
n³13311728219727443375
💡 Properties of Perfect Cubes

② Unit Digit of Cubes

Unlike squares, a cube can end in any digit (0 through 9). The unit digit of n³ depends only on the unit digit of n. If n ends in 2, n³ ends in 8. If n ends in 8, n³ ends in 2. Complementary pairs: (2,8), (3,7).

③ Even & Odd Rule

The cube of an even number is always even, and the cube of an odd number is always odd. Same rule as squares!

④ Sum of Consecutive Odd Numbers

Every perfect cube can be expressed as a sum of consecutive odd numbers.
1³ = 1
2³ = 3 + 5 = 8
3³ = 7 + 9 + 11 = 27
4³ = 13 + 15 + 17 + 19 = 64

⑤ Negative Cubes

Unlike squares, the cube of a negative number is negative. (−3)³ = −27. This is because (negative) × (negative) × (negative) = negative.
📊 Unit Digit Pattern for Cubes
Unit digit of n0123456789
Unit digit of n³0187456329
💡 Observation: Notice that 0, 1, 4, 5, 6, 9 map to themselves (unit digit of n³ = unit digit of n), while 2 ↔ 8 and 3 ↔ 7 swap. This is useful for quick checking!
🔸 Numbers That Are Both Perfect Squares and Cubes

Some special numbers are both perfect squares and perfect cubes. These are called perfect sixth powers because if n = a² = b³, then n = c6 for some c.

✅ 1 = 1² = 1³

Also 16 = 1

✅ 64 = 8² = 4³

Also 26 = 64

✅ 729 = 27² = 9³

Also 36 = 729

✅ 4096 = 64² = 16³

Also 46 = 4096
🚗 Taxicab Numbers

A taxicab number is the smallest number that can be expressed as the sum of two cubes in two or more different ways. These numbers are named after a famous story involving the great Indian mathematician Srinivasa Ramanujan.

📖 The Story: When the British mathematician G.H. Hardy visited Ramanujan in hospital, he mentioned that his taxi had the rather dull number 1729. Ramanujan immediately replied: "No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
1729 = 1³ + 12³ = 9³ + 10³ The Hardy-Ramanujan Number (first taxicab number)

Verification:

1³ + 12³ = 1 + 1728 = 1729 ✅
9³ + 10³ = 729 + 1000 = 1729 ✅
📊 More Taxicab Numbers
TaxicabNumberRepresentations
Ta(1)21³ + 1³ (only one way — trivial)
Ta(2)17291³ + 12³ = 9³ + 10³
Ta(3)87539319167³ + 436³ = 228³ + 423³ = 255³ + 414³
💡 Why is this special? It shows how deeply connected cube numbers are. Finding such numbers requires checking millions of combinations — yet Ramanujan spotted 1729's special property instantly from memory, showing his extraordinary genius with numbers.

🏆 Try It Yourself

Can you verify that there is no number smaller than 1729 that can be written as a sum of two cubes in two ways? Start by listing cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331... and check sums!

🔢 Cube Sums Near 1729

2³ + 12³ = 8 + 1728 = 1736
3³ + 12³ = 27 + 1728 = 1755
5³ + 11³ = 125 + 1331 = 1456
6³ + 11³ = 216 + 1331 = 1547
None of these match another cube sum!
📋 Finding Cube Roots

The cube root of a number n is the value m such that m × m × m = n. We write it as ³√n.

① Prime Factorization Method for Cube Roots
  1. Find the prime factorization of the number.
  2. Group the prime factors into triplets (groups of three).
  3. Take one factor from each triplet.
  4. Multiply the chosen factors — this is the cube root.

Example 4: Find ³√1728.

Step 1: Prime factorize 1728.
1728 = 2 × 864 = 2 × 2 × 432 = 2 × 2 × 2 × 216
= 2 × 2 × 2 × 2 × 108 = 2 × 2 × 2 × 2 × 2 × 54
= 2 × 2 × 2 × 2 × 2 × 2 × 27 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
1728 = 26 × 3³
Step 2: Group into triplets.
1728 = (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)
Step 3: Take one from each triplet.
³√1728 = 2 × 2 × 3 = 12
Verification: 12 × 12 × 12 = 1728 ✅

Example 5: Find ³√13824.

Step 1: Prime factorize 13824.
13824 = 2 × 6912 = 2² × 3456 = ... continuing...
13824 = 29 × 3³
Step 2: Group into triplets.
= (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)
Step 3: Take one from each triplet.
³√13824 = 2 × 2 × 2 × 3 = 24
Verification: 24 × 24 × 24 = 13824 ✅
🔧 Interactive: Cube Root Calculator
📦 Find the Cube Root
Enter a number to see if it is a perfect cube and find its cube root.
Enter a number above and click "Find ³√" to see the result.
💡 Key Difference:
• For square roots, group prime factors into pairs (groups of 2).
• For cube roots, group prime factors into triplets (groups of 3).
If any factor is left without a complete group, the number is NOT a perfect square/cube.
🔸 Pythagorean Triples from Squares

A Pythagorean triple is a set of three positive integers (a, b, c) such that a² + b² = c². These triples are deeply connected to perfect squares!

For any natural number m > 1:
(2m, m² − 1, m² + 1) is a Pythagorean triple This formula generates infinitely many Pythagorean triples
m2mm² − 1m² + 1TripleCheck: a² + b² = c²
2435(3, 4, 5)9 + 16 = 25 ✅
36810(6, 8, 10)36 + 64 = 100 ✅
481517(8, 15, 17)64 + 225 = 289 ✅
5102426(10, 24, 26)100 + 576 = 676 ✅
6123537(12, 35, 37)144 + 1225 = 1369 ✅
📚 Common Pythagorean Triples to Remember

🔴 (3, 4, 5)

The most basic triple.
3² + 4² = 9 + 16 = 25 = 5²

🟠 (5, 12, 13)

5² + 12² = 25 + 144 = 169 = 13²

🔵 (8, 15, 17)

8² + 15² = 64 + 225 = 289 = 17²

✅ (7, 24, 25)

7² + 24² = 49 + 576 = 625 = 25²
💡 Important: If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. For example, since (3, 4, 5) is a triple, so are (6, 8, 10), (9, 12, 15), (12, 16, 20), etc.

Example 6: Write a Pythagorean triple with one number as 20.

Method: Use the formula (2m, m² − 1, m² + 1).
If 2m = 20, then m = 10.
m² − 1 = 100 − 1 = 99
m² + 1 = 100 + 1 = 101
Answer: The Pythagorean triple is (20, 99, 101).
Verification: 20² + 99² = 400 + 9801 = 10201 = 101² ✅
✏️ NCERT-Style Worked Examples

Example 7: Find the smallest number by which 1152 must be multiplied to make it a perfect square.

Step 1: Prime factorize 1152.
1152 = 27 × 3²
Step 2: Check for pairs.
27 = (2 × 2) × (2 × 2) × (2 × 2) × 2 (one 2 is unpaired)
3² = (3 × 3) (complete pair)
Step 3: The factor 2 is unpaired. To complete the pair, we need one more 2.
Multiply by 2: 1152 × 2 = 2304 = 28 × 3²
Answer: Smallest number = 2. √2304 = 24 × 3 = 48.

Example 8: Find the smallest number by which 3600 must be divided to make it a perfect cube.

Step 1: Prime factorize 3600.
3600 = 24 × 3² × 5²
Step 2: Check for triplets.
24 = (2 × 2 × 2) × 2 (one 2 extra)
3² = 3 × 3 (two 3s, need one more for triplet)
5² = 5 × 5 (two 5s, need one more for triplet)
Step 3: To make it a perfect cube by division, we must remove the incomplete triplets.
Divide by 2¹ × 3² × 5² = 2 × 9 × 25 = 450
3600 ÷ 450 = 8 = 2³
Answer: Smallest number = 450. ³√8 = 2.

Example 9: Without calculating, determine whether 5929 is a perfect square.

Step 1: Check unit digit.
5929 ends in 9 — possible perfect square (unit digit is in {0,1,4,5,6,9}).
Step 2: Check digital root.
5 + 9 + 2 + 9 = 25 → 2 + 5 = 7 — this is in {1,4,7,9}. ✅ Still possible.
Step 3: Estimate.
76² = 5776, 77² = 5929. So 5929 = 77².
Answer: Yes, 5929 is a perfect square (√5929 = 77).

Example 10: Express 81 as a sum of consecutive odd numbers.

Method: Since 81 = 9², we need the sum of the first 9 odd numbers.
The first 9 odd numbers are: 1, 3, 5, 7, 9, 11, 13, 15, 17
Answer: 81 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17
Quick check: Sum = 9² = 81 ✅
📊 Chapter Summary
📋 All Formulas at a Glance
n² = n × n   |   n³ = n × n × n Definition of square and cube
1 + 3 + 5 + ... + (2n − 1) = n² Sum of first n odd numbers = n squared
(n + 1)² − n² = 2n + 1 Difference of consecutive squares
Pythagorean Triple: (2m, m² − 1, m² + 1) Generates triples for any m > 1
⚠️ Common Mistakes to Avoid

❌ (−5)² ≠ −25

(−5)² = (−5) × (−5) = +25. The square is always positive. But −5² = −(5²) = −25 (squaring applies to 5 only).

❌ Not All Odd Numbers Are Squares

The sum of first n odd numbers equals n², but individual odd numbers (like 7, 11, 13) are not perfect squares.

❌ Pairing vs Triplets

For square roots, group prime factors in pairs. For cube roots, group in triplets. Mixing these up is a common error.

❌ Forgetting to Check All Factors

When finding the smallest multiplier/divisor, every prime factor must be checked for complete pairs (or triplets).
📚 Key Terms Recap
TermDefinition
Perfect SquareA number that is the square of a natural number (1, 4, 9, 16, ...)
Perfect CubeA number that is the cube of a natural number (1, 8, 27, 64, ...)
Square Root (√n)The number whose square is n
Cube Root (³√n)The number whose cube is n
Pythagorean TripleThree positive integers (a, b, c) where a² + b² = c²
Prime FactorizationExpressing a number as a product of prime numbers
💡 Exam Day Checklist:
✅ Know the first 20 perfect squares and first 10 perfect cubes by heart.
✅ Unit digit test: squares never end in 2, 3, 7, or 8.
✅ Sum of first n odd numbers = n².
✅ For square roots: pair the prime factors. For cube roots: make triplets.
✅ Pythagorean triple formula: (2m, m² − 1, m² + 1).
🧠 Multiple Choice Questions (10 MCQs)

Test your understanding! Click on an option to check your answer.

  • Q1. Which of the following is a perfect square?
    • a) 222
    • b) 333
    • c) 225
    • d) 198
    ✅ Answer: (c) 225 — 225 = 15². Numbers 222 and 333 end in 2 and 3, so they cannot be perfect squares.
  • Q2. The square of an odd number is always:
    • a) Even
    • b) Odd
    • c) Prime
    • d) Divisible by 4
    ✅ Answer: (b) Odd — odd × odd = odd. For example, 7² = 49 (odd).
  • Q3. The cube root of 512 is:
    • a) 6
    • b) 7
    • c) 8
    • d) 9
    ✅ Answer: (c) 8 — 8 × 8 × 8 = 512.
  • Q4. 1 + 3 + 5 + 7 + 9 equals:
    • a) 20
    • b) 25
    • c) 30
    • d) 16
    ✅ Answer: (b) 25 — This is the sum of the first 5 odd numbers, which equals 5² = 25.
  • Q5. Which of the following numbers is NOT a perfect cube?
    • a) 125
    • b) 343
    • c) 200
    • d) 1000
    ✅ Answer: (c) 200 — 200 = 2³ × 5². Since 5 does not form a complete triplet, 200 is not a perfect cube.
  • Q6. The value of √(1 + 3 + 5 + 7 + 9 + 11 + 13) is:
    • a) 6
    • b) 7
    • c) 8
    • d) 49
    ✅ Answer: (b) 7 — Sum of first 7 odd numbers = 7² = 49, so √49 = 7.
  • Q7. (n + 1)² − n² is always equal to:
    • a) 2n
    • b) 2n + 1
    • c) n + 1
    • d) n² + 1
    ✅ Answer: (b) 2n + 1 — (n+1)² − n² = n² + 2n + 1 − n² = 2n + 1.
  • Q8. If 2m = 12, m² − 1 = 35, m² + 1 = 37, then (12, 35, 37) is a:
    • a) Fibonacci sequence
    • b) Pythagorean triple
    • c) Arithmetic progression
    • d) Geometric sequence
    ✅ Answer: (b) Pythagorean triple — 12² + 35² = 144 + 1225 = 1369 = 37².
  • Q9. The smallest number by which 72 must be multiplied to get a perfect cube is:
    • a) 2
    • b) 3
    • c) 6
    • d) 9
    ✅ Answer: (b) 3 — 72 = 2³ × 3². We need one more 3 to complete the triplet. 72 × 3 = 216 = 6³.
  • Q10. 35² can be quickly computed as:
    • a) 1125
    • b) 1025
    • c) 1225
    • d) 1325
    ✅ Answer: (c) 1225 — Using the shortcut: 3 × 4 = 12, append 25 → 1225.
✍️ NCERT Questions & Answers
📝 Fill in the Blanks
  • 1. The square of 13 is _______.
    169 (13 × 13 = 169)
  • 2. √256 = _______.
    16 (16 × 16 = 256)
  • 3. The cube of 7 is _______.
    343 (7 × 7 × 7 = 343)
  • 4. A perfect square can never end in the digits _______, _______, _______, or _______.
    2, 3, 7, or 8. Perfect squares can only end in 0, 1, 4, 5, 6, or 9.
  • 5. The sum of the first 8 odd numbers is _______.
    64 (Sum of first n odd numbers = n² = 8² = 64)
🔶 True or False
  • 1. The square of an even number is always even.
    True. Even × even = even. Example: 8² = 64 (even).
  • 2. 2000 is a perfect cube.
    False. 2000 = 24 × 5³. The factor 2 does not form a complete triplet (24 has one 2 extra), so 2000 is not a perfect cube.
  • 3. (−4)³ = −64.
    True. (−4) × (−4) × (−4) = 16 × (−4) = −64.
  • 4. There are 10 perfect squares between 1 and 100 (inclusive).
    True. They are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (that is 1² through 10²).
  • 5. 729 is both a perfect square and a perfect cube.
    True. 729 = 27² = 9³ = 36. It is a perfect sixth power.
✍️ Short Answer Questions
  • Q1. Using the pattern of consecutive odd numbers, find 6².
    6² = sum of first 6 odd numbers = 1 + 3 + 5 + 7 + 9 + 11 = 36.
  • Q2. Find √2025 by prime factorization.
    2025 = 34 × 5² = (3² × 5)² = 45².
    Therefore, √2025 = 45.
  • Q3. Is 392 a perfect cube? If not, find the smallest number by which it should be multiplied to make it a perfect cube.
    392 = 2³ × 7². The factor 7 appears only twice (needs a triplet). Multiply by 7 to get 392 × 7 = 2744 = 2³ × 7³ = 14³.
  • Q4. Find a Pythagorean triple in which one number is 14.
    Using the formula with 2m = 14, so m = 7.
    m² − 1 = 48, m² + 1 = 50.
    Triple: (14, 48, 50).
    Check: 14² + 48² = 196 + 2304 = 2500 = 50² ✅
  • Q5. Express 64 as (i) a perfect square, (ii) a perfect cube, and (iii) a sum of consecutive odd numbers.
    (i) 64 = 8²
    (ii) 64 = 4³
    (iii) As a sum (for 8²): 64 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 (sum of first 8 odd numbers)
    Or as a cube sum (for 4³): 64 = 13 + 15 + 17 + 19
📖 Long Answer Questions
Q1. Find the square root of 7744 by the long division method. Show all steps clearly.
Step 1: Group digits in pairs from right: 77 | 44.

Step 2: First group = 77. Largest square ≤ 77 is 64 (8²). First digit = 8.
Subtract: 77 − 64 = 13. Bring down 44 to get 1344.

Step 3: Double the quotient: 2 × 8 = 16_.
Find d such that 16d × d ≤ 1344.
Try d = 8: 168 × 8 = 1344. Exactly right!

Answer: √7744 = 88
Verification: 88 × 88 = 7744 ✅
Q2. Explain the relationship between perfect squares and the sum of consecutive odd numbers. Using this property, (a) express 144 as a sum of odd numbers, and (b) find the sum 1 + 3 + 5 + ... + 25.
Relationship: The sum of the first n consecutive odd numbers (starting from 1) always equals n². This is because each new odd number (2k − 1) added to the sum takes us from (k−1)² to k².

(a) Express 144 as a sum of odd numbers:
144 = 12², so it is the sum of the first 12 odd numbers:
144 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23

(b) Find 1 + 3 + 5 + ... + 25:
The nth odd number is 2n − 1. If 2n − 1 = 25, then n = 13.
So this is the sum of the first 13 odd numbers = 13² = 169.
Q3. The area of a square field is 5184 m². Find the perimeter of the field. Also, find the cost of fencing at Rs 25 per metre.
Step 1: Side of square = √5184.
5184 = 26 × 34 = (2³ × 3²)² = 72².
Side = 72 m.

Step 2: Perimeter = 4 × side = 4 × 72 = 288 m.

Step 3: Cost of fencing = 288 × 25 = Rs 7200.
Q4. A gardener wants to plant trees in rows and columns such that the number of rows equals the number of columns. He has 1024 trees. Can he plant all the trees? If so, how many rows will there be? If he gets 200 more trees, what is the smallest number of additional trees needed to maintain equal rows and columns?
Part 1: We need to check if 1024 is a perfect square.
1024 = 210 = (25)² = 32².
Yes! He can plant all 1024 trees in 32 rows and 32 columns.

Part 2: Total trees = 1024 + 200 = 1224.
The next perfect square after 1224 is 1225 = 35².
Additional trees needed = 1225 − 1224 = 1 more tree.
With 1225 trees, he plants in 35 rows × 35 columns.
Q5. Find the smallest perfect square that is divisible by each of 6, 9, and 15.
Step 1: Find LCM of 6, 9, and 15.
6 = 2 × 3,   9 = 3²,   15 = 3 × 5.
LCM = 2 × 3² × 5 = 90.

Step 2: Prime factorize 90 = 2 × 3² × 5.
For a perfect square, each factor must appear an even number of times.
2 appears once (need one more 2).
3 appears twice (already even).
5 appears once (need one more 5).

Step 3: Multiply by 2 × 5 = 10.
90 × 10 = 900.

Answer: 900 is the smallest perfect square divisible by 6, 9, and 15.
Verification: 900 = 30², and 900 ÷ 6 = 150 ✅, 900 ÷ 9 = 100 ✅, 900 ÷ 15 = 60 ✅.
🌟 Fun Facts & Did You Know?

💫 The Magic of 1089

33² = 1089 and 33³ = 35937. Now reverse 1089 to get 9801. And guess what? 99² = 9801! Also, 1089 × 9 = 9801. Numbers love symmetry!

🇮🇳 Ramanujan's Taxicab Number

The great Indian mathematician Srinivasa Ramanujan identified 1729 as the smallest number expressible as the sum of two cubes in two different ways: 1729 = 1³ + 12³ = 9³ + 10³. It is called the "Hardy-Ramanujan number."

🔢 Kaprekar's Squares

Indian mathematician D.R. Kaprekar discovered that some numbers, when squared and split into halves, add up to the original number. Example: 45² = 2025, and 20 + 25 = 45! Try 55² = 3025 → 30 + 25 = 55!

🚀 Cubes in Architecture

The Kaaba in Mecca is roughly cube-shaped, and its name literally means "cube" in Arabic. In modern architecture, the Apple Park building's floor plan uses geometric squares extensively.

🔬 Square Numbers in Nature

Sunflower seeds arrange themselves in spiral patterns related to square numbers and the Fibonacci sequence. The number of seeds often relates to perfect squares and golden ratios.

🎮 Rubik's Cube Maths

A standard Rubik's Cube has 3³ = 27 smaller cubes (26 visible). The total number of possible positions is approximately 4.3 × 1019 — more than there are grains of sand on Earth!
💡 Think About It: Squares and cubes are the building blocks of higher mathematics. Every formula in physics, every encryption algorithm in computers, and every geometric measurement relies on powers and roots. When you study squares and cubes, you are learning the language of the universe!
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