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A Square and a Cube

NCERT Class 8 · Mathematics Based on NCERT Class 8 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the concepts of square numbers, cube numbers, and their roots, using the engaging story of a puzzle involving 100 lockers to explain why only square-numbered lockers remain open. It explores patterns, properties, and methods to identify perfect squares and cubes, including prime factorisation and successive subtraction of odd numbers. The one big idea a student must walk away with is that a number is a perfect square if its prime factors can be split into two identical groups, and a perfect cube if split into three identical groups. This chapter builds foundational number sense for higher mathematics and appears frequently in board exams through direct questions on square roots, cube roots, and properties of squares and cubes.
Key Concepts (14)
Square number
A number that can be expressed as the product of a number with itself is called a square number, or simply a square.
e.g. 4 is a square number because 2 × 2 = 4. If you have a square garden of side 2 metres, its area is 4 square metres.
Perfect square
The squares of natural numbers are called perfect squares.
e.g. 1, 4, 9, 16, 25 are perfect squares because they are squares of natural numbers 1, 2, 3, 4, 5.
Square root
If y = x², then x is the square root of y. Every perfect square has two integer square roots: one positive and one negative.
e.g. √64 = ±8 because 8² = 64 and (-8)² = 64. In this chapter, we only consider the positive square root.
Cube number
A number obtained by multiplying a number by itself three times is called a cube.
e.g. 8 is a cube because 2 × 2 × 2 = 8. A Rubik's cube of side 2 units has 8 small cubes inside.
Perfect cube
A number that can be expressed as the product of a number with itself three times is called a perfect cube.
e.g. 1, 8, 27, 64, 125 are perfect cubes because they are 1³, 2³, 3³, 4³, 5³.
Cube root
If y = x³, then x is the cube root of y, denoted by ³√y.
e.g. ³√8 = 2 because 2³ = 8. ³√27 = 3 because 3³ = 27.
Factor
A factor of a number is a whole number that divides it exactly without leaving a remainder.
e.g. Factors of 6 are 1, 2, 3, and 6 because 6 ÷ 1 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2, 6 ÷ 6 = 1.
Partner factor
Each factor of a number has a 'partner factor' such that the product of the pair yields the given number.
e.g. For 6, 1 and 6 are partner factors (1×6=6), and 2 and 3 are partner factors (2×3=6).
Prime number
A number that has exactly two factors: 1 and the number itself.
e.g. 2, 3, 5, 7, 11 are prime numbers. Locker numbers that are prime are toggled exactly twice (by Person 1 and by themselves).
Triangular number
A number that can be represented as a triangular arrangement of dots, e.g., 1, 3, 6, 10, 15.
e.g. The sum of two consecutive triangular numbers gives a square number: 1 + 3 = 4 = 2², 3 + 6 = 9 = 3².
Taxicab number
A number that can be expressed as the sum of two cubes in two different ways.
e.g. 1729 = 1³ + 12³ = 9³ + 10³. It is the smallest taxicab number, also called the Hardy-Ramanujan number.
Varga
The ancient Sanskrit term used both for a square figure and for the square power of a number.
e.g. Aryabhata stated that the product of two equal quantities is called varga.
Ghana
The ancient Sanskrit term used both for a solid cube and for the product of a number with itself three times.
e.g. Ghana-mula was the term for cube-root in ancient Indian mathematics.
Mula
The Sanskrit word meaning root of a plant, basis, cause, or origin, used for the mathematical operation of taking roots.
e.g. Varga-mula means square-root, and ghana-mula means cube-root.
Formulas & Equations (7)
n × n = n²
Reads as: n squared equals n times n
Variables: n is any number
Use when: To find the square of a number, or to express the area of a square with side n.
⚠ Students write n² = n × 2 instead of n × n.
n × n × n = n³
Reads as: n cubed equals n times n times n
Variables: n is any number
Use when: To find the cube of a number, or to express the volume of a cube with edge n.
⚠ Students write n³ = n × 3 instead of n × n × n.
√y = x if x² = y
Reads as: square root of y equals x if x squared equals y
Variables: y is a perfect square, x is its positive square root
Use when: To find the side length of a square given its area, or to solve problems involving square roots.
⚠ Students forget the ± sign and write only the positive root, or they think √64 = 32.
³√y = x if x³ = y
Reads as: cube root of y equals x if x cubed equals y
Variables: y is a perfect cube, x is its cube root
Use when: To find the edge length of a cube given its volume, or to solve problems involving cube roots.
⚠ Students confuse cube root with square root, e.g., writing ³√8 = 4 instead of 2.
nth odd number = 2n – 1
Reads as: the nth odd number equals 2n minus 1
Variables: n is the position of the odd number in the sequence (1st, 2nd, 3rd, ...)
Use when: To find any odd number in sequence, e.g., the 36th odd number is 2×36 – 1 = 71.
⚠ Students use 2n + 1 instead of 2n – 1 for the nth odd number.
Sum of first n odd numbers = n²
Reads as: the sum of the first n odd numbers equals n squared
Variables: n is the number of odd terms added
Use when: To find a square by adding consecutive odd numbers starting from 1, or to check if a number is a perfect square.
⚠ Students think the sum of odd numbers from 1 to any odd number is always a square, but it only works starting from 1.
Difference between consecutive squares = next odd number
Reads as: the difference between two consecutive square numbers equals the next odd number
Variables: For squares n² and (n+1)², the difference is 2n+1
Use when: To find the next square from a given square, e.g., 36² = 35² + 71.
⚠ Students think the difference is always 2, but it increases by 2 each time.
How It Works
Determining if a locker is open or closed (the 100 locker puzzle)
  1. Step 1: Each locker is toggled by persons whose number is a factor of the locker number.
  2. Step 2: Count the number of factors of the locker number.
  3. Step 3: If the number of factors is odd, the locker is toggled an odd number of times and remains open.
  4. Step 4: If the number of factors is even, the locker is toggled an even number of times and remains closed.
  5. Step 5: Only square numbers have an odd number of factors, so only lockers with square numbers (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) remain open.
💡 Students forget that a number like 1 has only one factor (1×1), so it is a square and has an odd number of factors.
Checking if a number is a perfect square using prime factorisation
  1. Step 1: Write the prime factorisation of the number.
  2. Step 2: Group the prime factors into pairs of identical factors.
  3. Step 3: If all factors can be paired, the number is a perfect square.
  4. Step 4: The square root is the product of one factor from each pair.
  5. Step 5: Example: 324 = 2×2×3×3×3×3 = (2×3×3)² = 18², so √324 = 18.
💡 Students often forget to check if the factors can be split into two identical groups, not just any pairs.
Checking if a number is a perfect cube using prime factorisation
  1. Step 1: Write the prime factorisation of the number.
  2. Step 2: Group the prime factors into triplets of identical factors.
  3. Step 3: If all factors can be grouped into triplets, the number is a perfect cube.
  4. Step 4: The cube root is the product of one factor from each triplet.
  5. Step 5: Example: 3375 = 3×3×3×5×5×5 = (3×5)³ = 15³, so ³√3375 = 15.
💡 Students confuse the grouping for squares (pairs) with cubes (triplets) and may try to pair factors for cubes.
Finding square root by successive subtraction of odd numbers
  1. Step 1: Start with the given number.
  2. Step 2: Subtract consecutive odd numbers starting from 1 (1, 3, 5, 7, ...).
  3. Step 3: Continue until you reach exactly 0.
  4. Step 4: The number of subtractions performed is the square root.
  5. Step 5: Example: For 25, subtract 1→24, 3→21, 5→16, 7→9, 9→0. Five subtractions, so √25 = 5.
💡 If you never reach 0 and go negative, the number is not a perfect square. Students often stop before reaching 0.
Estimating square root of a non-perfect square
  1. Step 1: Find the nearest perfect squares below and above the given number.
  2. Step 2: Take the square roots of those perfect squares to get a range.
  3. Step 3: Narrow down by checking squares of numbers in between.
  4. Step 4: Example: For √250, 15²=225 and 16²=256, so √250 is between 15 and 16, closer to 16.
💡 Students forget to check which perfect square is closer to get a better estimate.
Finding cube root by estimation (without factorisation)
  1. Step 1: Look at the last digit of the cube to guess the last digit of the cube root.
  2. Step 2: Ignore the last three digits and look at the remaining number to find the tens digit.
  3. Step 3: Use the fact that cubes of 0-9 have unique last digits (0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9).
  4. Step 4: Example: For 1331, last digit 1 means cube root ends in 1, ignore '331', remaining 1, 1³=1, so cube root is 11.
💡 Students often confuse the mapping: e.g., 2³=8 so if a cube ends in 8, its cube root ends in 2, not 8.
Exam Traps — Don't Make These Mistakes
⚠ Confusing square and cube notation
✗ n² = n × 2 and n³ = n × 3
→ The exponent indicates how many times the base is multiplied by itself, not multiplied by the exponent. n² means n × n, not n × 2.
✓ n² = n × n and n³ = n × n × n
⚠ Forgetting the ± sign for square roots
✗ √64 = 8
→ Every perfect square has two square roots: a positive and a negative one, because (-8)² = 64 also. The question may ask for both.
✓ √64 = ±8
⚠ Assuming a number ending in 6 is a perfect square
✗ 26 is a perfect square because it ends in 6.
→ While perfect squares can only end in 0, 1, 4, 5, 6, or 9, not every number ending in these digits is a square. 26 ends in 6 but is not a square.
✓ A number ending in 2, 3, 7, or 8 is definitely not a square, but ending in 0,1,4,5,6,9 does not guarantee it is a square.
⚠ Mistaking the number of zeros in a square
✗ If a number ends with 3 zeros, its square ends with 3 zeros.
→ The number of zeros at the end of a square is double the number of zeros at the end of the original number. For example, 1000 has 3 zeros, its square 1,000,000 has 6 zeros.
✓ Squares can only have an even number of zeros at the end. If a number has 3 zeros, its square has 6 zeros.
⚠ Thinking all odd numbers have odd cubes
✗ The cube of any odd number is odd.
→ This statement is actually true, but the trap is that students often say 'the cube of any odd number is even' (which is false). The correct statement: odd × odd × odd = odd.
✓ The cube of an odd number is always odd. The cube of an even number is always even.
⚠ Confusing factors for squares vs cubes
✗ A number is a perfect square if its prime factors can be split into three identical groups.
→ That is the condition for a perfect cube, not a perfect square. For a perfect square, factors must be split into two identical groups.
✓ A number is a perfect square if its prime factors can be split into two identical groups. A number is a perfect cube if its prime factors can be split into three identical groups.
Compare & Contrast
Definition (multiplication)
Square number
Cube number
A square number is obtained by multiplying a number by itself once (n × n = n²), while a cube number is obtained by multiplying a number by itself twice (n × n × n = n³).
Inverse operation
Square root
Cube root
Square root is the inverse of squaring: if x² = y, then √y = x (two possible values: ±x). Cube root is the inverse of cubing: if x³ = y, then ³√y = x (only one real value, which can be positive or negative).
Grouping of prime factors
Perfect square condition (prime factorisation)
Perfect cube condition (prime factorisation)
For a perfect square, prime factors must be grouped into pairs (two identical groups). For a perfect cube, prime factors must be grouped into triplets (three identical groups).
Possible units digits
Last digit of a square
Last digit of a cube
Perfect squares can only end in 0, 1, 4, 5, 6, or 9. Perfect cubes can end in any digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) depending on the number.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Very Short Answer (1 mark), 1 Short Answer (2-3 marks), 1 Long Answer (4-5 marks). Total approximately 8-10 marks.
Most asked: Identifying perfect squares and perfect cubes using prime factorisation · Finding square roots and cube roots by prime factorisation · Properties of squares (units digit, number of zeros, odd/even) · Difference between consecutive squares · Sum of first n odd numbers equals n² · Estimating square roots
Don't skip: Finding the smallest number to multiply or divide to make a number a perfect square or perfect cube; Word problems involving area of square (square root) or volume of cube (cube root); Prime factorisation method for square root and cube root
One-Line Revision (read aloud before the exam)
  1. A square number is obtained by multiplying a number by itself: n × n = n².
  2. A perfect square is the square of a natural number, e.g., 1, 4, 9, 16, 25.
  3. A cube number is obtained by multiplying a number by itself three times: n × n × n = n³.
  4. A perfect cube is the cube of a natural number, e.g., 1, 8, 27, 64, 125.
  5. Only square numbers have an odd number of factors.
  6. Perfect squares end only in 0, 1, 4, 5, 6, or 9; never in 2, 3, 7, or 8.
  7. Squares can only have an even number of zeros at the end.
  8. The sum of the first n odd numbers equals n².
  9. The nth odd number is given by 2n – 1.
  10. The difference between consecutive squares (n+1)² – n² = 2n + 1.
  11. A number is a perfect square if its prime factors can be split into two identical groups.
  12. A number is a perfect cube if its prime factors can be split into three identical groups.
  13. Every perfect square has two square roots: a positive and a negative.
  14. The cube root of a number is the inverse of cubing it.
  15. 1729 is the smallest taxicab number, expressible as 1³+12³ and 9³+10³.
  16. To estimate a square root, find the nearest perfect squares above and below the number.
  17. If a number ends with 2, 3, 7, or 8, it is definitely not a perfect square.

MCQs — 10 Questions with Answers

Q1. Which of the following is a perfect square?

  • A. 36 ✓
  • B. 48
  • C. 52
  • D. 68

Answer: A — 36 = 6 × 6 = 6², while 48, 52, and 68 cannot be expressed as the product of a number with itself.

Q2. How many factors does the number 16 have?

  • A. 2
  • B. 3
  • C. 4
  • D. 5 ✓

Answer: D — The factors of 16 are 1, 2, 4, 8, and 16 — a total of 5 factors, which is odd, confirming 16 is a perfect square.

Q3. Which digit can a perfect square NOT end with?

  • A. 0
  • B. 5
  • C. 7 ✓
  • D. 9

Answer: C — Perfect squares can only end in 0, 1, 4, 5, 6, or 9; the digit 7 never appears in the units place of any perfect square.

Q4. If locker number n is toggled by every person whose number divides n evenly, which lockers remain open?

  • A. All lockers
  • B. Lockers with prime numbers
  • C. Lockers with perfect square numbers ✓
  • D. Lockers with even numbers

Answer: C — A locker remains open if toggled an odd number of times, which happens only for perfect squares because they have an unpaired factor.

Q5. At a market, tiles are arranged to form a square display. If there are 64 tiles total, how many tiles are on each side?

  • A. 6
  • B. 7
  • C. 8 ✓
  • D. 9

Answer: C — Since 8 × 8 = 64, or 8² = 64, each side of the square display has 8 tiles.

Q6. What is the sum 1 + 3 + 5 + 7 + 9 + 11 equal to?

  • A. 36 ✓
  • B. 49
  • C. 64
  • D. 81

Answer: A — The sum of the first 6 consecutive odd numbers equals 6² = 36, demonstrating that consecutive odd numbers from 1 sum to perfect squares.

Q7. A farmer has a square field with an area of 225 square meters. What is the length of one side of the field?

  • A. 12 meters
  • B. 13 meters
  • C. 14 meters
  • D. 15 meters ✓

Answer: D — If area = 225 m², then side length = √225 = 15 m, since 15 × 15 = 225.

Q8. Which method would correctly identify whether 144 is a perfect square?

  • A. Check if it ends in 0, 1, 4, 5, 6, or 9
  • B. Subtract consecutive odd numbers until reaching 0 or going negative
  • C. Find if it has an even number of factors
  • D. Both B and C ✓

Answer: D — Both methods work: 144 ends in 4 (passes test A), has 15 factors which is odd (method C), and subtracting odd numbers reaches 0 at step 12 (method B).

Q9. If you know that 20² = 400, what can you deduce about 200² without detailed calculation?

  • A. 200² = 40,000 because 200 = 10 × 20 ✓
  • B. 200² = 4,000 because there are two zeros
  • C. 200² = 400,000 because the exponent applies to both digits
  • D. Cannot be determined without calculating

Answer: A — Since 200 = 10 × 20, then (200)² = (10 × 20)² = 100 × 400 = 40,000, showing the multiplication principle for squares of multiples.

Q10. Using the pattern that the nth odd number is 2n - 1, what is the 15th odd number?

  • A. 27
  • B. 29 ✓
  • C. 31
  • D. 33

Answer: B — The 15th odd number = 2(15) - 1 = 30 - 1 = 29; this formula allows finding any odd number's position without listing all previous ones.

Flashcards

What is a perfect square?

A number that can be expressed as the product of a number with itself, like 4 = 2 × 2 or 9 = 3 × 3.

Why does only locker number 36 toggle an odd number of times?

Because 36 = 6 × 6, and the factor 6 pairs with itself, making the total number of factors odd (1, 2, 3, 4, 6, 9, 12, 18, 36).

Which digits can never appear in the units place of a perfect square?

The digits 2, 3, 7, and 8 never appear in the units place of any perfect square.

What is the relationship between consecutive odd numbers and perfect squares?

The sum of the first n consecutive odd numbers starting from 1 equals n²; for example, 1 + 3 + 5 + 7 = 16 = 4².

Write the formula for the nth odd number.

The nth odd number is 2n - 1, so the 10th odd number is 2(10) - 1 = 19.

If a number has two zeros at the end, how many zeros will its square have?

Four zeros, because (20)² = 400, demonstrating that the number of zeros in a square is always even and doubles the count in the original number.

What does the symbol √ mean?

It represents the positive square root of a number; for example, √64 = 8 because 8 × 8 = 64.

How can you check if 48 is a perfect square without calculating?

Subtract consecutive odd numbers starting from 1 until you reach zero or a negative number; if you go negative without reaching zero, it is not a perfect square.

What is the only even prime number, and what is its square?

The number 2 is the only even prime number, and its square is 2² = 4.

If 35² = 1225, how can you find 36² using the odd number pattern?

Add the 36th odd number (which is 2(36) - 1 = 71) to 1225 to get 1296 = 36².

Important Board Questions

Is 144 a perfect square? Give one reason. [1 mark]

Check if 144 equals the product of a number with itself (12 × 12 = 144), OR check if it has an odd number of factors, OR check if it ends in an allowed digit.

Explain why the number 250 cannot be a perfect square using the units digit test. [2 marks]

Identify the units digit (0 is allowed), so this test doesn't eliminate it. Instead, use: perfect squares ending in 0 must have an EVEN number of zeros. 250 has one zero (odd), so it cannot be a perfect square.

Find 25² using the property that perfect squares equal the sum of consecutive odd numbers. Show your working step by step. [3 marks]

25² = sum of the first 25 odd numbers. List the pattern: 1 + 3 + 5 + ... and add them. Alternatively, use 24² = 576 (sum of first 24 odd numbers), then add the 25th odd number (which is 2(25) - 1 = 49) to get 576 + 49 = 625.

A school principal plans to arrange 196 students in a perfect square formation for the annual sports day parade. (a) Is this possible? (b) How many students will stand in each row? (c) Explain your answer by finding the square root and verifying with factor pairs. [5 marks]

Check if 196 is a perfect square: (a) Find √196 by testing if 14² works (14 × 14 = 196), OR list factor pairs of 196 and count (1 × 196, 2 × 98, 4 × 49, 7 × 28, 14 × 14). (b) The answer is 14 students per row. (c) Since 14 × 14 = 196, there are 15 factors total (odd count), confirming it is a perfect square. Show the formation as a 14 × 14 grid.

Assertion-Reason Questions for A Square and a Cube

Q1. Assertion (A):

The number 144 has an odd number of factors.

Reason (R):

144 = 12 × 12 is a perfect square, and every perfect square has an odd number of factors.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both statements are true. 144 is indeed a perfect square (12²), and the reason correctly explains why it has an odd number of factors. The unpaired factor is 12 itself (12 × 12 = 144). All perfect squares have exactly one factor that pairs with itself, giving an odd total count of factors.

Q2. Assertion (A):

Every number that ends in 5 is a perfect square.

Reason (R):

Numbers like 5, 15, 25, 35 can all end in 5.

Answer: D — A is false but R is true — The assertion is FALSE. While 25 = 5² and 225 = 15² are perfect squares ending in 5, numbers like 5, 15, 35, 45, 55, 65, 75, 85, 95 end in 5 but are NOT perfect squares. The reason is irrelevant and merely states that numbers ending in 5 exist, which does not support the assertion. A units digit of 5 is necessary but not sufficient for a number to be a perfect square.

Q3. Assertion (A):

If we sum the first n consecutive odd numbers starting from 1, we always get a perfect square.

Reason (R):

The sum of the first n odd numbers equals n², which is a perfect square by definition.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason IS the correct explanation. The mathematical relationship 1 + 3 + 5 + ... + (2n−1) = n² is demonstrated in the chapter through pattern observation and visual proof with L-shapes. The reason directly explains why the assertion is true.

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Case-Based Questions (CBQ) for A Square and a Cube

Case Study:

A school is planning to redesign its playground by creating square activity zones. The principal wants to arrange square zones in a grid pattern. For a 6 × 6 grid of zones, she needs to calculate the total number of zones, the number of zones on the border (perimeter), and predict how many zones she would need for a 10 × 10 grid. She also wants to know which grid sizes will produce a perfect square number of zones that might correspond to perfect square areas when each zone is 1 meter × 1 meter.

Q1. How many zones are in the 6 × 6 grid?

  • A. 36 ✓
  • B. 12
  • C. 30
  • D. 42

Answer: A

Q2. If each zone is a 1 m × 1 m square, what is the total area of all 36 zones in the 6 × 6 grid?

  • A. 6 m²
  • B. 36 m² ✓
  • C. 72 m²
  • D. 30 m²

Answer: B

Q3. How many zones will a 10 × 10 grid have?

  • A. 100 ✓
  • B. 20
  • C. 50
  • D. 120

Answer: A

Q4. Which of the following grid sizes will create a perfect square number of zones?

  • A. 7 × 7
  • B. 8 × 8
  • C. Both 7 × 7 and 8 × 8 ✓
  • D. Neither 7 × 7 nor 8 × 8

Answer: C

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