1.5 Going Deeper: Enrichment & Exam Preparation

Where Squares and Cubes Show Up in Real Life

Math in the Real World

  • Tiling and floor plans. A square hall of side 9 m needs 9^2 = 81 square metres of tiles. Areas are always measured in squared units, which is exactly why we call the operation “squaring.”
  • Packing and volume. A storage cube of side 4 cm holds 4^3 = 64 unit cubes. Volumes use cubed units for the same reason.
  • The square–cube law. Double the side of a cube and its surface area grows \times 4 (a square effect) but its volume grows \times 8 (a cube effect). This single fact explains why a mouse can survive a fall that would injure an elephant, and why a giant shaped exactly like a person could not stand up, their weight (volume) would outrun the strength of their bones (cross-sectional area).
  • Everyday squares and cubes. A chessboard has 8^2 = 64 squares; a Rubik’s cube is built from 3^3 = 27 small cubes; a 1080\times 1080 profile picture is a perfect square of pixels.
  • A bridge to later chapters. Squares power the Pythagoras theorem a^2 + b^2 = c^2 (Chapter 9), and repeated multiplication grows into exponents and scientific notation (Chapter 2).

Quick Reference: Squares and Cubes

Knowing these by heart turns slow calculations into instant recall, a real advantage under exam time pressure.

n n^2 n n^2 n n^2
1 1 11 121 21 441
2 4 12 144 22 484
3 9 13 169 23 529
4 16 14 196 24 576
5 25 15 225 25 625
6 36 16 256 26 676
7 49 17 289 27 729
8 64 18 324 28 784
9 81 19 361 29 841
10 100 20 400 30 900
n n^3 n n^3
1 1 11 1331
2 8 12 1728
3 27 13 2197
4 64 14 2744
5 125 15 3375
6 216 16 4096
7 343 17 4913
8 512 18 5832
9 729 19 6859
10 1000 20 8000

Use the explorer to generate the squares and cubes for any range you want to practise:

Squares & cubes explorer

Memory Tricks & One-Page Revision

Quick Revision Card

  • Last digit of a square is always one of 0,1,4,5,6,9, never 2,3,7,8.
  • Squaring a number ending in 5: a5 \to a(a+1) then write 25. (E.g. 75^2: 7\times 8 = 56 \to 5625.)
  • Last digit of a cube: same digit, except 2\leftrightarrow 8 and 3\leftrightarrow 7. This lets you read off a cube root’s units digit at sight.
  • Is it a perfect square? Pair up the prime factors, every prime must appear an even number of times.
  • Is it a perfect cube? Group the prime factors in threes, every prime must appear a multiple-of-3 number of times.
  • Roots: \sqrt{\ } gives the positive root; x^2=k gives \pm\sqrt{k}. The cube root keeps the sign: \sqrt[3]{-27} = -3.
  • Sum of first n odd numbers = n^2; the nth cube is a run of n consecutive odd numbers.

Spot the Mistake

Common Exam Mistakes

  • Writing x = 7 only when solving x^2 = 49, the full answer is x = \pm 7.
  • Assuming any number ending in 6 is a square. 26, 46, 76 all end in 6 yet none is a perfect square.
  • Confusing “the square of 9” (9^2 = 81) with “the square root of 9” (\sqrt{9} = 3).
  • Thinking a perfect square can end in an odd number of zeros, it cannot; squares end in an even count of zeros.
  • Forgetting that the cube root keeps the sign of a negative number: \sqrt[3]{-64} = -4, not +4.

Test Your Reflexes

A fast, friendly drill: a perfect square or cube appears, and you type its root. Build a streak!

Root reflex
Type the root and press Check.
√144 = ?
Score 0 · Streak 0

Exam Corner: CBSE-Style Practice

Mixed Practice (objective, short, long, HOTS)

Objective type (1 mark each)

  1. The units digit of 93^2 is ______.
  2. The value of \sqrt{2.25} is ______.
  3. How many whole numbers lie between 20^2 and 21^2?

Short answer (2 marks each)

  1. Find the least number that must be added to 525 to make it a perfect square.
  2. Find \sqrt{7056} using prime factorisation.

Long answer (3 marks each)

  1. Find the smallest number by which 8748 must be divided to obtain a perfect cube, and state the cube root of the result.
  2. A gardener has 1000 saplings to plant in a square arrangement with an equal number of rows and columns. Find the least number of extra saplings needed so that none are left over, and state how many rows there will be.

HOTS (Higher Order Thinking)

  1. Find every digit d for which d and d^3 end in the same units digit.
  2. Show that no three-digit number with all three digits identical (such as 111, 222, \dots, 999) can be a perfect square.

Assertion–Reason (Choose: (a) both true, R explains A; (b) both true, R does not explain A; (c) A true, R false; (d) A false, R true.)

  1. Assertion (A): 1728 is a perfect cube.   Reason (R): A number is a perfect cube when its prime factors can be grouped into identical triples.
  1. 93^2 = 8649, so the units digit is 9. (Quick check: a number ending in 3 has a square ending in 9.)
  2. \sqrt{2.25} = \mathbf{1.5}, since 1.5^2 = 2.25.
  3. Between 20^2 and 21^2 there are 2\times 20 = \mathbf{40} whole numbers.
  4. The next square after 525 is 23^2 = 529, so add 529 - 525 = \mathbf{4}.
  5. 7056 = 2^4 \times 3^2 \times 7^2 = (2^2\times 3\times 7)^2 = 84^2, so \sqrt{7056} = \mathbf{84}.
  6. 8748 = 2^2 \times 3^7. For a cube each exponent must be a multiple of 3: remove 2^2 and one 3, i.e. divide by 2^2\times 3 = \mathbf{12}. Then 8748\div 12 = 729 = 9^3, so the cube root is \mathbf{9}.
  7. 31^2 = 961 and 32^2 = 1024. Since 1000 lies between them, add 1024 - 1000 = \mathbf{24} saplings to make 1024 = 32^2, giving 32 rows (and 32 columns).
  8. Testing d = 0,1,\dots,9, the cube ends in the same digit exactly for d \in \{\mathbf{0,1,4,5,6,9}\}. (For 2,3,7,8 the last digit changes, the swaps 2\leftrightarrow 8 and 3\leftrightarrow 7.)
  9. A repeated-digit number is d \times 111 = d \times 3 \times 37. For a perfect square the prime 37 would need a partner, but it appears only once, so the number can never be a perfect square. (Checking 111,222,\dots,999 directly confirms none is a square.)
  10. (a), Both are true and R is the correct explanation: 1728 = 2^6\times 3^3 = (2^2\times 3)^3 = 12^3, a perfect cube exactly because its prime factors form identical triples.

Connections to Other Chapters

How This Chapter Links Forward

  • Chapter 2 (Power Play): squaring and cubing are the first steps into the world of exponents, n^2 and n^3 being powers with index 2 and 3.
  • Chapter 9 (Baudhayana–Pythagoras Theorem): the relation a^2 + b^2 = c^2 runs entirely on squares and square roots.
  • Chapter 11 & 14 (Solids and Area): areas live in squared units and volumes in cubed units, the same idea that named these operations.

Glossary

Key Terms

  • Square number / perfect square: a number equal to some integer times itself, e.g. 25 = 5^2.
  • Square root (\sqrt{\ }): the inverse of squaring; the positive value whose square is the number.
  • Radical sign: the symbol \sqrt{\ } (and \sqrt[3]{\ } for cube root).
  • Cube number / perfect cube: a number equal to some integer times itself three times, e.g. 27 = 3^3.
  • Cube root (\sqrt[3]{\ }): the inverse of cubing; keeps the sign of the original number.
  • Prime factorisation: writing a number as a product of primes, the key to testing for squares (pairs) and cubes (triples).
  • Parity: whether a number is even or odd; squaring and cubing both preserve parity.