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Class 8 Ganita Prakash · Chapter 1
Class 8 Ganita Prakash · Chapter 1
A Square and A Cube
Page-wise textbook solutions (pages 1–18) · Every step · Figures · Common mistakes · Tips
๐️ Key ideas of this chapter
- A square is a number times itself: \(n \times n = n^2\). A cube is a number times itself three times: \(n \times n \times n = n^3\).
- Squares end only in 0, 1, 4, 5, 6 or 9, and have an even number of zeros at the end.
- Sum of the first \(n\) odd numbers \(= n^2\). There are \(2n\) numbers between \(n^2\) and \((n+1)^2\).
- Perfect square → prime factors make pairs. Perfect cube → prime factors make triplets.
- \(\sqrt{\;}\) means positive square root; \(\sqrt[3]{\;}\) means cube root.
๐ Textbook Page 1
๐ Page 1Locker puzzle
100 lockers are closed. Person 1 opens every locker. Person 2 changes (toggles) every 2nd locker, Person 3 every 3rd locker, and so on up to Person 100. Khoisnam knew before they started which lockers would be open at the end. How did he know?
1Locker number 6 is touched by Persons 1, 2, 3 and 6 — exactly the factors of 6.
2So, the number of times a locker is toggled = the number of factors of the locker number.
3Toggling goes open → closed → open → closed … So a locker ends open only if it is toggled an odd number of times.
4Factors come in partner pairs: \(6 = 1 \times 6 = 2 \times 3\). Pairs give an even count of factors.
5But in \(9 = 3 \times 3\), the factor 3 is its own partner, so it is counted only once. Now the count is odd (1, 3, 9).
6This happens only for numbers that are a number times itself — the square numbers.
✅ Answer: Only lockers with square numbers stay open: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
⚠️ Common mistake: Thinking every number has factors in pairs. A square number has one factor that pairs with itself.
๐ก Tip: Odd number of toggles = OPEN. Even number of toggles = CLOSED.
๐ Textbook Page 2
๐ Page 2Think
Does every number have an even number of factors?
1Check a few numbers.
21 → factor: 1 (only one factor).
34 → factors: 1, 2, 4 (three factors).
49 → factors: 1, 3, 9 (three factors).
✅ Answer: No. Square numbers like 1, 4, 9 have an odd number of factors.
๐ Page 2Think
In \(4 = 2 \times 2\), both numbers in the pair are the same. Use this idea to find more numbers with an odd number of factors.
1We need a pair where a number is multiplied by itself: \(n \times n\).
2\(16 = 4 \times 4\) → factors 1, 2, 4, 8, 16 → 5 factors (odd).
3\(25 = 5 \times 5\) → factors 1, 5, 25 → 3 factors (odd).
4\(49 = 7 \times 7\) → factors 1, 7, 49 → 3 factors (odd).
✅ Answer: 16, 25, 36, 49, 64, … — all square numbers have an odd number of factors.
๐ Page 2Check
36 has the pair 6 × 6. Check that every other factor of 36 has a different partner. Does 36 have an odd number of factors?
1Write all the factor pairs of 36:
2\(1 \times 36,\quad 2 \times 18,\quad 3 \times 12,\quad 4 \times 9,\quad 6 \times 6\)
3Every factor except 6 has a different partner.
4Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36.
5Count = 9, which is odd.
✅ Answer: Yes. 36 has 9 factors — an odd number.
⚠️ Common mistake: Writing 6 two times in the list of factors. List each factor only once.
๐ Textbook Page 3
๐ Page 3Locker puzzle
Write the numbers of the lockers that stay open.
1The open lockers are the square numbers from 1 to 100.
2\(1^2=1,\;2^2=4,\;3^2=9,\;4^2=16,\;5^2=25\)
3\(6^2=36,\;7^2=49,\;8^2=64,\;9^2=81,\;10^2=100\)
✅ Answer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (10 lockers).
๐ Page 3Locker puzzle
The clue says: the passcode is the first five locker numbers that were touched exactly twice. Which are these lockers?
1A locker is touched exactly twice if its number has exactly two factors.
2Numbers with exactly two factors (1 and itself) are prime numbers.
3The first five primes are 2, 3, 5, 7, 11.
✅ Answer: Passcode: 2 – 3 – 5 – 7 – 11.
⚠️ Common mistake: Including 1. The number 1 has only one factor, so locker 1 is touched only once. 1 is not prime.
๐ Page 3Think
Can we have a square with side \(\tfrac{3}{5}\) unit or 2.5 units? What are their areas?
1Area of a square = side × side. The side does not need to be a whole number.
2\(\left(\tfrac{3}{5}\right)^2 = \tfrac{3}{5} \times \tfrac{3}{5} = \tfrac{9}{25}\) sq. unit
3\((2.5)^2 = 2.5 \times 2.5 = 6.25\) sq. units
✅ Answer: Yes. Areas are \(\tfrac{9}{25}\) sq. unit and 6.25 sq. units.
๐ก Tip: For a decimal, square the number without the point (25 × 25 = 625) and then put back double the decimal places (6.25).
๐ Textbook Page 4
๐ Page 4Table
Find the squares of the first 30 natural numbers and complete the table.
1Multiply each number by itself.
2Shortcut: \((n+1)^2 = n^2 + n + (n+1)\). For example \(13^2 = 144 + 12 + 13 = 169\).
3The completed table is below.
| 1 to 10 | 11 to 20 | 21 to 30 |
|---|---|---|
| \(1^2 = 1\) | \(11^2 = 121\) | \(21^2 = 441\) |
| \(2^2 = 4\) | \(12^2 = 144\) | \(22^2 = 484\) |
| \(3^2 = 9\) | \(13^2 = 169\) | \(23^2 = 529\) |
| \(4^2 = 16\) | \(14^2 = 196\) | \(24^2 = 576\) |
| \(5^2 = 25\) | \(15^2 = 225\) | \(25^2 = 625\) |
| \(6^2 = 36\) | \(16^2 = 256\) | \(26^2 = 676\) |
| \(7^2 = 49\) | \(17^2 = 289\) | \(27^2 = 729\) |
| \(8^2 = 64\) | \(18^2 = 324\) | \(28^2 = 784\) |
| \(9^2 = 81\) | \(19^2 = 361\) | \(29^2 = 841\) |
| \(10^2 = 100\) | \(20^2 = 400\) | \(30^2 = 900\) |
✅ Answer: See the completed table.
๐ก Tip: Learn the squares up to 30 by heart. They save a lot of time in exams.
๐ Page 4Math Talk
Study the table. What patterns do you notice?
1Squares end only in 0, 1, 4, 5, 6 or 9 — never in 2, 3, 7 or 8.
2Square of an even number is even; square of an odd number is odd.
3Numbers ending in 1 or 9 → square ends in 1. Ending in 2 or 8 → ends in 4. Ending in 3 or 7 → ends in 9. Ending in 4 or 6 → ends in 6. Ending in 5 → ends in 25.
4The gap between two squares in a row is an odd number: 3, 5, 7, 9, …
| Number ends in | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Square ends in | 0 | 1 | 4 | 9 | 6 | 5 | 6 | 9 | 4 | 1 |
✅ Answer: Many patterns — the units digit pattern is the most useful one.
๐ Page 4Math Talk
If a number ends in 0, 1, 4, 5, 6 or 9, is it always a square?
1Try examples: 26 ends in 6 but is not a square (5² = 25, 6² = 36).
210, 11, 14, 15, 19 also end in these digits but are not squares.
3So the units digit can only tell us when a number is NOT a square.
✅ Answer: No. Example: 26 ends in 6 but is not a square.
⚠️ Common mistake: Saying a number is a square just because it ends in 1, 4, 5, 6, 9 or 0. Always check further.
๐ Page 4Practice
Write 5 numbers which you can tell are not squares just by looking at the units digit.
1Any number ending in 2, 3, 7 or 8 is not a square.
2Examples: 52, 123, 2047, 998, 3333.
✅ Answer: 52, 123, 2047, 998, 3333 (any numbers ending in 2, 3, 7 or 8).
๐ Page 4Practice
\(1^2, 9^2, 11^2, 19^2, 21^2, 29^2\) all end in 1. Write the next two squares that end in 1.
1A square ends in 1 when the number ends in 1 or 9.
2After 29, the next such numbers are 31 and 39.
3\(31^2 = 961\)
4\(39^2 = 1521\)
✅ Answer: \(31^2 = 961\) and \(39^2 = 1521\).
๐ Textbook Page 5
๐ Page 5Practice
Which of these squares have 6 in the units place?
(i) \(38^2\) (ii) \(34^2\) (iii) \(46^2\) (iv) \(56^2\) (v) \(74^2\) (vi) \(82^2\)
(i) \(38^2\) (ii) \(34^2\) (iii) \(46^2\) (iv) \(56^2\) (v) \(74^2\) (vi) \(82^2\)
1Look only at the units digit and square it.
2(i) \(8 \times 8 = 64\) → ends in 4
3(ii) \(4 \times 4 = 16\) → ends in 6
4(iii) \(6 \times 6 = 36\) → ends in 6
5(iv) \(6 \times 6 = 36\) → ends in 6
6(v) \(4 \times 4 = 16\) → ends in 6
7(vi) \(2 \times 2 = 4\) → ends in 4
✅ Answer: (ii) \(34^2\), (iii) \(46^2\), (iv) \(56^2\), (v) \(74^2\).
๐ก Tip: If a number ends in 4 or 6, its square ends in 6.
๐ Page 5Think
If a number has 3 zeros at the end, how many zeros will its square have at the end?
1Example: \(2000^2 = 2000 \times 2000\).
2\(2 \times 2 = 4\), and the zeros add up: 3 + 3 = 6 zeros.
3\(2000^2 = 4\,000\,000\)
✅ Answer: 6 zeros.
๐ Page 5Think
What do you notice about the zeros at the end of a number and of its square? Will it always happen? Can squares have only an even number of zeros at the end?
1\(10^2 = 100\): 1 zero → 2 zeros. \(700^2 = 490000\): 2 zeros → 4 zeros.
2When we multiply the number by itself, the zeros at the end get doubled.
3Double of any whole number is even. So the square always has 0, 2, 4, 6, … zeros at the end.
✅ Answer: The square has twice as many end-zeros. Yes, always. So a square can only have an even number of zeros at the end (e.g. 1000 is not a square).
⚠️ Common mistake: Thinking 1000 or 40 000 000 can be a square. Count the zeros: an odd count means NOT a square.
๐ Page 5Think
What can you say about the parity (even or odd) of a number and its square?
1Even × even = even. So the square of an even number is even. Example: 6² = 36.
2Odd × odd = odd. So the square of an odd number is odd. Example: 7² = 49.
✅ Answer: A number and its square have the same parity.
๐ Page 5Pattern
4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, 25 − 16 = 9. Does this pattern continue?
136 − 25 = 11
249 − 36 = 13
364 − 49 = 15
4The differences are the odd numbers in order.
✅ Answer: Yes. Consecutive squares differ by consecutive odd numbers. So 1 + 3 + 5 + … (first n odd numbers) = n².
๐ Textbook Page 6
๐ Page 6Visual proof
Explain the picture: why does each new inverted L-shape give the next odd number?
1To grow an \(n \times n\) square into an \((n+1) \times (n+1)\) square, we add one row of \(n\), one column of \(n\), and 1 corner square.
2Squares added = \(n + n + 1 = 2n + 1\), which is always odd.
3So the L-shapes add 1, 3, 5, 7, 9, …
✅ Answer: Each L adds the next odd number, so the first n odd numbers make an n × n square.
๐ Page 6Worked
Given \(35^2 = 1225\), find \(36^2\) using the odd-number pattern.
11225 is the sum of the first 35 odd numbers.
2To get \(36^2\), add the 36th odd number.
3The nth odd number is \(2n - 1\), so the 36th odd number is \(2 \times 36 - 1 = 71\).
4\(36^2 = 1225 + 71 = 1296\)
✅ Answer: \(36^2 = 1296\).
๐ก Tip: Next square = this square + this number + next number: 1225 + 35 + 36 = 1296.
๐ Page 6Think
What is the nth odd number?
11st = 1, 2nd = 3, 3rd = 5, 6th = 11.
2Each one is 1 less than double the position: \(2 \times 6 - 1 = 11\).
✅ Answer: \(2n - 1\)
๐ Page 6Worked
Is 38 a perfect square? Use subtraction of odd numbers.
138 − 1 = 37, 37 − 3 = 34, 34 − 5 = 29, 29 − 7 = 22, 22 − 9 = 13, 13 − 11 = 2
22 − 13 goes below 0. We never land exactly on 0.
✅ Answer: No, 38 is not a perfect square.
๐ Textbook Page 7
๐ Page 7Pattern
How many numbers lie between two consecutive perfect squares? Do you see a pattern?
1Between 1 and 4: 2, 3 → 2 numbers.
2Between 4 and 9: 5, 6, 7, 8 → 4 numbers.
3Between 9 and 16: 10 to 15 → 6 numbers.
4In general, between \(n^2\) and \((n+1)^2\): \((n+1)^2 - n^2 - 1 = 2n\) numbers.
✅ Answer: There are \(2n\) numbers between \(n^2\) and \((n+1)^2\).
⚠️ Common mistake: Forgetting to subtract 1 — the difference 2n+1 counts one end square too.
๐ Page 7Table
How many squares are in each block of 100 up to 1000? What is the largest square less than 1000?
11–100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 → 10
2101–200: 121, 144, 169, 196 → 4
3201–300: 225, 256, 289 → 3
4301–400: 324, 361, 400 → 3
5401–500: 441, 484 → 2; 501–600: 529, 576 → 2; 601–700: 625, 676 → 2
6701–800: 729, 784 → 2; 801–900: 841, 900 → 2; 901–1000: 961 → 1
7\(31^2 = 961\) and \(32^2 = 1024\), which is more than 1000.
| 1–100 | 101–200 | 201–300 | 301–400 | 401–500 |
| 10 | 4 | 3 | 3 | 2 |
| 501–600 | 601–700 | 701–800 | 801–900 | 901–1000 |
| 2 | 2 | 2 | 2 | 1 |
✅ Answer: Largest square below 1000 is 961 (= 31²). Squares get fewer in each block as numbers grow.
๐ Page 7Pattern
What is the link between triangular numbers and square numbers? Draw the next term.
1Triangular numbers: 1, 3, 6, 10, 15, …
2Two triangular numbers next to each other add to a square:
31 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16
4Next term: 10 + 15 = 25 = 5²
✅ Answer: Sum of two consecutive triangular numbers is a square. Next term: 10 + 15 = 25 = 5².
๐ Textbook Page 8
๐ Page 8Think
What is the square root of 64?
1\(8 \times 8 = 64\) and \((-8) \times (-8) = 64\).
2So 64 has two square roots: +8 and −8.
3The symbol \(\sqrt{64}\) means the positive root.
✅ Answer: \(\sqrt{64} = 8\) (the two square roots are ±8).
⚠️ Common mistake: Forgetting that (−8) × (−8) is also 64 (minus × minus = plus).
๐ Page 8Worked
Is 576 a perfect square? (Method 1: list squares)
1\(20^2 = 400,\; 21^2 = 441,\; 22^2 = 484,\; 23^2 = 529,\; 24^2 = 576\)
✅ Answer: Yes, \(576 = 24^2\), so \(\sqrt{576} = 24\).
๐ Page 8Worked
Find \(\sqrt{81}\) by subtracting odd numbers.
181 − 1 = 80, 80 − 3 = 77, 77 − 5 = 72, 72 − 7 = 65, 65 − 9 = 56
256 − 11 = 45, 45 − 13 = 32, 32 − 15 = 17, 17 − 17 = 0
3We reached 0 in 9 steps.
✅ Answer: \(\sqrt{81} = 9\).
๐ก Tip: Number of subtractions = square root. Good for small numbers; slow for big ones like 729 (27 steps).
๐ Textbook Page 9
๐ Page 9Worked
Is 324 a perfect square? Use prime factorisation.
1\(324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3\)
2Make pairs: \((2 \times 2) \times (3 \times 3) \times (3 \times 3)\) — every factor has a partner.
3Take one from each pair: \(2 \times 3 \times 3 = 18\).
✅ Answer: Yes. \(\sqrt{324} = 18\).
๐ Page 9Worked
Is 156 a perfect square?
1\(156 = 2 \times 2 \times 3 \times 13\)
23 and 13 have no partners.
✅ Answer: No, 156 is not a perfect square.
๐ Page 9Practice
Using prime factorisation, find whether 1156 and 2800 are perfect squares.
1\(1156 = 2 \times 2 \times 17 \times 17\)
2Both factors are in pairs → perfect square.
3\(\sqrt{1156} = 2 \times 17 = 34\)
4\(2800 = 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 7\)
57 has no partner → not a perfect square.
✅ Answer: 1156 is a perfect square (\(34^2\)). 2800 is not.
๐ก Tip: Quick check for 2800: it ends in two zeros (OK), but 28 is not a square, so 2800 is not a square.
๐ Link ahead: In Class 9–10 you will use the same pairing idea to simplify surds like \(\sqrt{2800} = 20\sqrt{7}\).
๐ Page 9Worked
Find \(\sqrt{1936}\) by estimating.
1\(40^2 = 1600\) and \(50^2 = 2500\), so \(\sqrt{1936}\) is between 40 and 50.
21936 ends in 6, so the root ends in 4 or 6 → 44 or 46.
3\(45^2 = (40+5)^2 = 1600 + 400 + 25 = 2025\)
4\(2025 \gt 1936\), so the root is less than 45 → it is 44.
5Check: \(44 \times 44 = 1936\) ✔
✅ Answer: \(\sqrt{1936} = 44\).
๐ Textbook Page 10
๐ Page 10Worked
Estimate \(\sqrt{250}\).
1\(15^2 = 225\) and \(16^2 = 256\). So \(15 \lt \sqrt{250} \lt 16\).
2250 is much closer to 256 than to 225.
✅ Answer: \(\sqrt{250}\) is about 16 (a little less than 16; actually ≈ 15.8).
๐ Page 10Worked
Akhil has a square cloth of area 125 cm². Can he cut a square handkerchief of side 15 cm? If not, what is the biggest one with a whole-number side?
1\(15^2 = 225\) cm², which is more than 125 cm². So, No.
2\(11^2 = 121 \le 125\) and \(12^2 = 144 \gt 125\).
✅ Answer: No. The largest handkerchief has side 11 cm.
๐ Real life: Tailors, tilers and carpenters use this kind of estimate all the time.
๐ Page 10Figure it Out · Q1
Which of these are not perfect squares? (i) 2032 (ii) 2048 (iii) 1027 (iv) 1089
1(i) 2032 ends in 2 → not a square.
2(ii) 2048 ends in 8 → not a square.
3(iii) 1027 ends in 7 → not a square.
4(iv) 1089 ends in 9; check: \(30^2 = 900\), \(33^2 = 1089\) ✔ → a square.
✅ Answer: (i) 2032, (ii) 2048 and (iii) 1027 are not perfect squares.
๐ Page 10Figure it Out · Q2
Which among \(64^2, 108^2, 292^2, 36^2\) has last digit 4?
1\(64^2\): 4 × 4 = 16 → 6
2\(108^2\): 8 × 8 = 64 → 4
3\(292^2\): 2 × 2 = 4 → 4
4\(36^2\): 6 × 6 = 36 → 6
✅ Answer: \(108^2\) and \(292^2\).
๐ก Tip: Numbers ending in 2 or 8 give squares ending in 4.
๐ Page 10Figure it Out · Q3
Given \(125^2 = 15625\), what is \(126^2\)?
(i) 15625 + 126 (ii) 15625 + 262 (iii) 15625 + 253 (iv) 15625 + 251 (v) 15625 + 512
(i) 15625 + 126 (ii) 15625 + 262 (iii) 15625 + 253 (iv) 15625 + 251 (v) 15625 + 512
1\(126^2 = 125^2 + 125 + 126\) (add one strip of 125 and one strip of 126).
2125 + 126 = 251
3\(126^2 = 15625 + 251 = 15876\)
4Another way: 126th odd number = 2 × 126 − 1 = 251.
✅ Answer: (iv) 15625 + 251 = 15876.
⚠️ Common mistake: Choosing (i) 15625 + 126. You must add both 125 and 126.
๐ Page 10Figure it Out · Q4
Find the side of a square whose area is 441 m².
1Side = \(\sqrt{441}\).
2\(441 = 3 \times 3 \times 7 \times 7\)
3\(\sqrt{441} = 3 \times 7 = 21\)
✅ Answer: Side = 21 m.
⚠️ Common mistake: Writing the unit as m². Side is a length, so the unit is m.
๐ Page 10Figure it Out · Q5
Find the smallest square number that is divisible by each of 4, 9 and 10.
1The number must be a multiple of the LCM of 4, 9 and 10.
2\(4 = 2 \times 2,\; 9 = 3 \times 3,\; 10 = 2 \times 5\)
3LCM \(= 2 \times 2 \times 3 \times 3 \times 5 = 180\)
42s are paired, 3s are paired, but 5 is alone.
5Multiply by 5: \(180 \times 5 = 900 = 30^2\).
✅ Answer: 900
⚠️ Common mistake: Stopping at 180. 180 is divisible by all three, but it is not a square.
๐ Page 10Figure it Out · Q6
Find the smallest number by which 9408 must be multiplied to get a perfect square. Find the square root of the product.
1\(9408 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 7 \times 7\)
22s: three pairs ✔, 7s: one pair ✔, 3: alone ✘
3Multiply by 3.
4\(9408 \times 3 = 28224\)
5\(\sqrt{28224} = 2 \times 2 \times 2 \times 3 \times 7 = 168\)
✅ Answer: Multiply by 3. Product = 28224, square root = 168.
๐ก Tip: Divide by 2 again and again first: 9408 → 4704 → 2352 → 1176 → 588 → 294 → 147 = 3 × 7 × 7.
๐ Page 10Figure it Out · Q7
How many numbers lie between the squares of (i) 16 and 17 (ii) 99 and 100?
1Between \(n^2\) and \((n+1)^2\) there are \(2n\) numbers.
2(i) n = 16 → 2 × 16 = 32. (Check: 256 and 289 → 288 − 256 = 32.)
3(ii) n = 99 → 2 × 99 = 198.
✅ Answer: (i) 32 (ii) 198
๐ Textbook Page 11
๐ Page 11Figure it Out · Q8
Fill in the missing numbers:
\(1^2 + 2^2 + 2^2 = 3^2\)
\(2^2 + 3^2 + 6^2 = 7^2\)
\(3^2 + 4^2 + 12^2 = 13^2\)
\(4^2 + 5^2 + 20^2 = (\_\_)^2\)
\(9^2 + 10^2 + (\_\_)^2 = (\_\_)^2\)
\(1^2 + 2^2 + 2^2 = 3^2\)
\(2^2 + 3^2 + 6^2 = 7^2\)
\(3^2 + 4^2 + 12^2 = 13^2\)
\(4^2 + 5^2 + 20^2 = (\_\_)^2\)
\(9^2 + 10^2 + (\_\_)^2 = (\_\_)^2\)
1The third number = product of the first two: 1×2 = 2, 2×3 = 6, 3×4 = 12.
2The answer = third number + 1: 3, 7, 13.
3So \(4 \times 5 = 20\) and \(20 + 1 = 21\). Check: \(16 + 25 + 400 = 441 = 21^2\) ✔
4\(9 \times 10 = 90\) and \(90 + 1 = 91\). Check: \(81 + 100 + 8100 = 8281 = 91^2\) ✔
✅ Answer: \(4^2 + 5^2 + 20^2 = 21^2\) and \(9^2 + 10^2 + 90^2 = 91^2\).
๐ Link ahead: The rule is \(n^2 + (n+1)^2 + \big(n(n+1)\big)^2 = \big(n(n+1)+1\big)^2\) — you can prove it with algebra in Class 9.
๐ Page 11Figure it Out · Q9
How many tiny squares are in the picture (a big green square made of tilted and straight tiles)? Write the prime factorisation of this number.
1Each tile (straight or tilted) is a 5 × 5 grid = 25 tiny squares.
2The tiles are in 9 rows and 9 columns = 81 tiles.
3Tiny squares = \(81 \times 25 = 2025\)
4\(2025 = 3 \times 3 \times 3 \times 3 \times 5 \times 5 = 3^4 \times 5^2\)
5Also \(2025 = 45^2\) — a perfect square!
✅ Answer: 2025 tiny squares; \(2025 = 3^4 \times 5^2\).
๐ก Tip: Count one tile carefully, then count tiles — do not count tiny squares one by one.
๐ Page 11Think
How many cubes of side 1 cm make a cube of side 2 cm? And a cube of side 3 cm?
1Side 2 cm: 2 layers, each layer 2 × 2 = 4 cubes → 2 × 4 = 8.
2Side 3 cm: 3 layers, each layer 3 × 3 = 9 cubes → 3 × 9 = 27.
✅ Answer: 8 cubes (\(2^3\)) and 27 cubes (\(3^3\)).
๐ Textbook Page 12
๐ Page 12Think
Is 9 a perfect cube?
1\(2^3 = 8\) and \(3^3 = 27\).
29 lies between 8 and 27, so it is not a cube. No number from 9 to 26 is a cube.
✅ Answer: No.
๐ Page 12Table
Complete the table of cubes from 1 to 20.
1\(n^3 = n \times n \times n\). Example: \(12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728\).
2The completed table is below.
| 1 to 10 | 11 to 20 |
|---|---|
| \(1^3 = 1\) | \(11^3 = 1331\) |
| \(2^3 = 8\) | \(12^3 = 1728\) |
| \(3^3 = 27\) | \(13^3 = 2197\) |
| \(4^3 = 64\) | \(14^3 = 2744\) |
| \(5^3 = 125\) | \(15^3 = 3375\) |
| \(6^3 = 216\) | \(16^3 = 4096\) |
| \(7^3 = 343\) | \(17^3 = 4913\) |
| \(8^3 = 512\) | \(18^3 = 5832\) |
| \(9^3 = 729\) | \(19^3 = 6859\) |
| \(10^3 = 1000\) | \(20^3 = 8000\) |
✅ Answer: See the table.
๐ Page 12Math Talk
What patterns do you notice in the table of cubes?
1Cube of an even number is even; cube of an odd number is odd.
21 → 1, 4 → 4, 5 → 5, 6 → 6, 9 → 9, 0 → 0 (same units digit).
32 ↔ 8 and 3 ↔ 7 swap: 2³ ends in 8, 8³ ends in 2; 3³ ends in 7, 7³ ends in 3.
4Cubes grow very fast: 10³ = 1000, 20³ = 8000.
✅ Answer: Each units digit gives a different units digit in the cube.
๐ Textbook Page 13
๐ Page 13Think
Squares can only end in 0, 1, 4, 5, 6, 9. What are the possible last digits of cubes?
1From the table, the units digits of cubes are 1, 8, 7, 4, 5, 6, 3, 2, 9, 0.
2All ten digits appear.
| Number ends in | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Cube ends in | 0 | 1 | 8 | 7 | 4 | 5 | 6 | 3 | 2 | 9 |
✅ Answer: A cube can end in any digit 0 to 9.
๐ก Tip: So the units digit can NEVER tell you that a number is not a cube. But it tells you the units digit of the cube root!
๐ Page 13Think
How many cubes have 1 digit, 2 digits and 3 digits? What do you observe?
11 digit: 1, 8 → 2 cubes
22 digits: 27, 64 → 2 cubes
33 digits: 125, 216, 343, 512, 729 → 5 cubes
4(For comparison, there are 22 three-digit squares.)
✅ Answer: 2, 2 and 5. Cubes are much rarer than squares — they spread out quickly.
๐ Page 13Explain
Can a cube end with exactly two zeros (00)?
1If a number ends with k zeros, its cube ends with 3k zeros.
2\(10^3 = 1000\) (3 zeros), \(100^3 = 1\,000\,000\) (6 zeros).
3So cubes end with 0, 3, 6, 9, … zeros — never exactly 2.
✅ Answer: No. The number of end-zeros of a cube is always a multiple of 3.
๐ Page 13Try This
Taxicab numbers: write 4104 and 13832 each as the sum of two cubes in two ways.
1\(4104 = 2^3 + 16^3 = 8 + 4096\)
2\(4104 = 9^3 + 15^3 = 729 + 3375\)
3\(13832 = 2^3 + 24^3 = 8 + 13824\)
4\(13832 = 18^3 + 20^3 = 5832 + 8000\)
✅ Answer: \(4104 = 2^3+16^3 = 9^3+15^3\); \(13832 = 2^3+24^3 = 18^3+20^3\).
๐ Story: 1729 = 1³ + 12³ = 9³ + 10³ is the famous Hardy–Ramanujan number.
๐ Textbook Page 14
๐ Page 14Think
1 = 1³, 3 + 5 = 2³, 7 + 9 + 11 = 3³, … Without adding, find 91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109.
1Row 1 has 1 odd number, row 2 has 2, row 3 has 3 … Row n adds to n³.
2Count the numbers in this sum: 91, 93, …, 109 → 10 numbers.
3So it is row 10.
4Sum \(= 10^3 = 1000\).
✅ Answer: 1000
๐ก Tip: Check the first number: row n starts at n × (n − 1) + 1. For n = 10: 90 + 1 = 91 ✔
๐ Page 14Worked
Is 3375 a perfect cube? Is 500 a perfect cube?
1\(3375 = 3 \times 3 \times 3 \times 5 \times 5 \times 5 = (3 \times 5)^3 = 15^3\) → cube.
2\(500 = 2 \times 2 \times 5 \times 5 \times 5\) → the 2s do not make a triplet → not a cube.
✅ Answer: \(\sqrt[3]{3375} = 15\). 500 is not a perfect cube.
๐ Textbook Page 15
๐ Page 15Practice
Find (i) \(\sqrt[3]{64}\) (ii) \(\sqrt[3]{512}\) (iii) \(\sqrt[3]{729}\)
1(i) \(64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = (2 \times 2)^3 = 4^3\) → 4
2(ii) \(512 = 2^9 = (2 \times 2 \times 2)^3 = 8^3\) → 8
3(iii) \(729 = 3^6 = (3 \times 3)^3 = 9^3\) → 9
✅ Answer: (i) 4 (ii) 8 (iii) 9
⚠️ Common mistake: Making pairs instead of triplets. For cube roots, group in threes.
๐ Page 15Pattern
Find successive differences of the cubes 1, 8, 27, 64, 125, 216, … until all differences are the same. What do you notice?
1Level 1: 8 − 1 = 7, 27 − 8 = 19, 64 − 27 = 37, 125 − 64 = 61, 216 − 125 = 91
2Level 2: 12, 18, 24, 30 (go up by 6)
3Level 3: 6, 6, 6, … all the same
Cubes182764125216343
Level 1719376191127
Level 21218243036
Level 36666
✅ Answer: For cubes all differences become equal (6) at Level 3. For squares it happened at Level 2 (all 2).
๐ Link ahead: Squares settle at level 2, cubes at level 3, 4th powers at level 4. This idea is used in Class 11 sequences.
๐ Textbook Page 16
๐ Page 16Figure it Out · Q1
Find the cube roots of 27000 and 10648.
1\(27000 = 27 \times 1000 = 3^3 \times 10^3\) → \(\sqrt[3]{27000} = 3 \times 10 = 30\)
2\(10648 = 2 \times 2 \times 2 \times 11 \times 11 \times 11 = 2^3 \times 11^3\)
3\(\sqrt[3]{10648} = 2 \times 11 = 22\)
✅ Answer: 30 and 22
๐ Page 16Figure it Out · Q2
What number will you multiply 1323 by to make it a cube?
1\(1323 = 3 \times 3 \times 3 \times 7 \times 7\)
23s make a triplet ✔. 7 appears only twice — one more 7 is needed.
3\(1323 \times 7 = 9261 = 21^3\)
✅ Answer: Multiply by 7 (product 9261 = 21³).
๐ Page 16Figure it Out · Q3
True or false? Give reasons.
(i) The cube of any odd number is even.
(ii) No perfect cube ends with 8.
(iii) The cube of a 2-digit number may be a 3-digit number.
(iv) The cube of a 2-digit number may have seven or more digits.
(v) Cube numbers have an odd number of factors.
(i) The cube of any odd number is even.
(ii) No perfect cube ends with 8.
(iii) The cube of a 2-digit number may be a 3-digit number.
(iv) The cube of a 2-digit number may have seven or more digits.
(v) Cube numbers have an odd number of factors.
1(i) False — odd × odd × odd is odd. Example: 3³ = 27.
2(ii) False — 2³ = 8 and 12³ = 1728 end in 8.
3(iii) False — the smallest 2-digit number is 10 and 10³ = 1000 already has 4 digits.
4(iv) False — the largest is 99³ = 970299, which has only 6 digits.
5(v) False — 8 has factors 1, 2, 4, 8 → 4 factors (even). (Only cubes that are also squares, like 1 and 64, have an odd number.)
✅ Answer: All five statements are false.
๐ก Tip: One counter-example is enough to prove a statement false.
๐ Page 16Figure it Out · Q4
1331 is a perfect cube. Guess its cube root without factorising. Also guess the cube roots of 4913, 12167 and 32768.
1Step A — units digit: look at the table (1→1, 3→7, 7→3, 8→2).
2Step B — tens digit: remove the last three digits and find the biggest cube ≤ what is left.
31331: ends in 1 → root ends in 1; left part 1 → \(1^3 \le 1\) → tens digit 1 → 11
44913: ends in 3 → root ends in 7; left part 4 → \(1^3 \le 4 \lt 2^3\) → 17
512167: ends in 7 → root ends in 3; left part 12 → \(2^3 \le 12 \lt 3^3\) → 23
632768: ends in 8 → root ends in 2; left part 32 → \(3^3 \le 32 \lt 4^3\) → 32
✅ Answer: 11, 17, 23, 32
⚠️ Common mistake: Mixing up 3 and 7 (or 2 and 8). A cube ending in 3 has a root ending in 7.
๐ Textbook Page 17
๐ Page 17Figure it Out · Q5
Which is the greatest? (i) \(67^3 - 66^3\) (ii) \(43^3 - 42^3\) (iii) \(67^2 - 66^2\) (iv) \(43^2 - 42^2\)
1Difference of consecutive squares: \(67^2 - 66^2 = 67 + 66 = 133\); \(43^2 - 42^2 = 85\).
2Difference of consecutive cubes: \(n^3 - (n-1)^3 = 3n(n-1) + 1\).
3\(67^3 - 66^3 = 3 \times 67 \times 66 + 1 = 13267\)
4\(43^3 - 42^3 = 3 \times 43 \times 42 + 1 = 5419\)
5Cubes grow faster than squares, and bigger numbers have bigger gaps.
✅ Answer: (i) \(67^3 - 66^3 = 13267\) is the greatest.
๐ Textbook Page 18
๐ Page 18Square Pairs! (Puzzle)
Arrange 1 to 17 in a row so that every two neighbours add up to a square. Is there more than one way? Then try 1 to 32 in a circle.
1Possible square sums: 4, 9, 16, 25 (the biggest sum is 16 + 17 = 33).
216 can only go with 9 (16 + 9 = 25). 17 can only go with 8 (17 + 8 = 25).
3A number with only one partner must sit at an end. So 16 and 17 are the two ends.
4Build from 16 and follow the only possible choices:
516, 9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8, 17
6Since both ends are forced, this is the only way (reading it backwards is the same row).
7For 1 to 32 in a circle, every number needs two partners. One answer is shown in the circle below.
16259167921614251116594161225131639616102515161982517
✅ Answer: Row: 16-9-7-2-14-11-5-4-12-13-3-6-10-15-1-8-17 (only one way, apart from reversing).
๐ก Tip: Start with the numbers that have the fewest partners — they decide the rest.
๐ Where do we use squares and cubes?
- Area of floors, tiles, plots and screens (square units); volume of boxes, tanks and rooms (cubic units).
- Estimating square roots helps in cutting cloth, laying tiles and fencing.
- Scientists and engineers use squares and cubes in speed, energy and design formulas.
๐ Link to higher classes
- Class 9: real numbers, surds such as \(\sqrt{2}\), algebraic identities like \((a+b)^2\).
- Class 10: surface area and volume, quadratic equations.
- Class 11: sequences and the sum of squares and cubes.
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