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CBSE · Class 8 · Mathematics Part 1

Part 1, Chapter 1: A Square and A Cube

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Square numbers and their patterns

From daily life

A tile-fitter lays 7 rows of 7 tiles in a hall corner. The tiles needed are 7 × 7 = 49, a square number.

Simple definition

The square of a number is the number multiplied by itself: n × n = n² (read "n squared"). Numbers such as 1, 4, 9, 16, 25 are perfect squares because that many dots make a full square.

1 + 3 + 5 + 7 + 9= 25 = 5²each L-shape = next odd number

In the figure: Each coloured L-shape adds the next odd number: 1, 3, 5, 7, 9. So the sum of the first n odd numbers is n².

Rule 1: Look at the last digit — A perfect square ends only in 0, 1, 4, 5, 6 or 9. If a number ends in 2, 3, 7 or 8, it is surely not a square.

  • Easy: 27² ends in 9, because 7 × 7 = 49.
  • Medium: 12² = 144 and 38² = 1444: numbers ending in 2 or 8 give squares ending in 4.
  • Exam-type: Is 5372 a square? It ends in 2, so NO.

Rule 2: Count the zeros — If a number ends in n zeros, its square ends in 2n zeros. So a square always has an even number of zeros at the end.

  • Easy: 60² = 3600: one zero became two zeros.
  • Medium: 500² = 250000: two zeros became four.
  • Exam-type: Can 72000 be a square? It has 3 zeros (odd), so NO.

Rule 3: Use odd numbers — (n + 1)² = n² + (2n + 1). To go to the next square, add the next odd number.

  • Easy: 1 + 3 + 5 + 7 + 9 + 11 + 13 = 7² = 49 (seven odd numbers).
  • Medium: Given 19² = 361, then 20² = 361 + 39 = 400 (39 = 2 × 19 + 1).
  • Exam-type: 1 + 3 + 5 + … (30 terms) = 30² = 900, and the last term is 2 × 30 − 1 = 59.
Remember

A square has an odd number of factors, and (odd)² is odd, (even)² is even.

Common mistake

Thinking that every number ending in 1, 4, 5, 6, 9 or 0 is a square. 26 ends in 6 but is not a square; the last digit can only rule a number OUT.

Shortcut

To square a number ending in 5, multiply the tens part by the next number and write 25. For 75²: 7 × 8 = 56, so 5625. Memory clue: Squares finish with "0 1 4 5 6 9" - say "zero-one-four-five-six-nine, only six can finish a square". One more odd number builds the next square.

Spot the difference

PointSquare of an ODD numberSquare of an EVEN number
Is the squareoddeven
Example9² = 818² = 64
Possible last digits1, 5, 90, 4, 6

Practice questions

Easy
  1. 1What is 24²?
  2. 2Is 4,938 a perfect square? Why?
  3. 3Add 1 + 3 + 5 + … (10 terms).
Medium
  1. 4Find 85² using the rule for numbers ending in 5.
  2. 5How many zeros are at the end of 3000²?
  3. 6Given 29² = 841, find 30² without multiplying.
Challenging
  1. 7A square can end in 1. Which numbers, ending in what digits, give such a square?
  2. 8The difference of the squares of two consecutive numbers is 83. Find the numbers.
Show answers (try them yourself first)
  1. 1576
  2. 2No: it ends in 8, and no square ends in 8
  3. 3100
  4. 47225 (8 × 9 = 72, then 25)
  5. 56
  6. 6841 + 59 = 900
  7. 7Numbers ending in 1 or 9 (e.g. 11² = 121, 19² = 361)
  8. 841 and 42 (41 + 42 = 83)

Square roots

From daily life

A square rangoli covers 144 square feet. To draw its border we must know the side. The side is the number whose square is 144: it is 12 feet.

Simple definition

If y = x², then x is the square root of y, written x = √y. A perfect square has two roots, +x and −x (for example +9 and −9 for 81). In this chapter we write only the positive root.

Live figure

Square roots

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Live figure

Square roots

In the app you can move this figure with your finger.

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Rule 1: Pair the prime factors — Write the prime factors, make pairs of equal factors and take ONE factor from each pair. Multiply them.

  • Easy: 144 = (2 × 2) × (2 × 2) × (3 × 3), so √144 = 2 × 2 × 3 = 12.
  • Medium: 1296 = 2⁴ × 3⁴, so √1296 = 2² × 3² = 4 × 9 = 36.
  • Exam-type: 5184 = 2⁶ × 3⁴, so √5184 = 2³ × 3² = 8 × 9 = 72.

Rule 2: Estimate between two squares — Find the two perfect squares around the number, then use the last digit to decide the root.

  • Easy: 49 < 50 < 64, so 7 < √50 < 8; and √50 is close to 7.
  • Medium: √2116: 1600 < 2116 < 2500, so the root is 40-something. 2116 ends in 6, so the root ends in 4 or 6: try 46, and 46² = 2116.
  • Exam-type: A cloth has area 300 cm². The largest square with whole-number side: 17² = 289 ≤ 300 < 324 = 18², so the side is 17 cm.
Remember

Repeated subtraction also works: 36 − 1 − 3 − 5 − 7 − 9 − 11 = 0 after 6 steps, so √36 = 6. And √(n²) = n, but also (−n)² = n².

Common mistake

Taking the square root by halving: √144 is NOT 72. Check by squaring: 72² is much bigger than 144.

Shortcut

Last digit of the root: number ends in 1 → root ends in 1 or 9; ends in 4 → 2 or 8; ends in 9 → 3 or 7; ends in 6 → 4 or 6; ends in 5 → 5. Memory clue: "Pair, then pick one": make pairs of equal factors and pick one from every pair (Marathi: जोडीतून एक). Check the answer by squaring it.

Spot the difference

PointPerfect squareNot a perfect square
Prime factorsall in pairssome are left single
Its root isa whole numberbetween two whole numbers
Example√196 = 14√200 is between 14 and 15

Practice questions

Easy
  1. 1Find √169.
  2. 2A square plot has area 256 m². Find its side.
  3. 3Between which two whole numbers does √70 lie?
Medium
  1. 4Find √900 using prime factors.
  2. 5Find √3136 (ends in 6; between 50 and 60).
  3. 6Find √(49/121).
Challenging
  1. 7A hall has 784 chairs in a square arrangement. How many chairs are in each row, and how many rows?
  2. 8A square has area 500 cm². What is the largest whole-number side of a square you can cut from it, and how much area is left over?
Show answers (try them yourself first)
  1. 113
  2. 216 m
  3. 38 and 9 (64 < 70 < 81)
  4. 430 (900 = 2² × 3² × 5²)
  5. 556 (56² = 3136)
  6. 67/11
  7. 728 chairs in a row, 28 rows (28² = 784)
  8. 8Side 22 cm (22² = 484); left over 500 − 484 = 16 cm²

Perfect squares: making and counting

From daily life

A shop has 98 identical tiles but wants a square floor with equal rows and columns. Should it order more tiles or return some? Prime factors help: 98 = 2 × 7 × 7, so 98 × 2 = 196 = 14² tiles make a perfect square.

Simple definition

A number is a perfect square when every prime factor appears an even number of times. The factor that has no partner decides what to multiply by or divide by.

6² = 367² = 4912 numbers between = 2 × 6

In the figure: Between 6² = 36 and 7² = 49 there are 2 × 6 = 12 non-square numbers.

Rule 1: Multiply to make a square — Multiply by the product of the unpaired prime factors.

Live figure

Perfect squares: making and counting

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  • Easy: 18 = 2 × 3 × 3. The 2 is single, so 18 × 2 = 36 = 6².
  • Medium: 252 = 2² × 3² × 7. Only 7 is single, so 252 × 7 = 1764 = 42².
  • Exam-type: 1008 = 2⁴ × 3² × 7. Multiply by 7: 7056 = 84².

Rule 2: Divide to make a square — Divide by the product of the unpaired prime factors.

  • Easy: 50 = 2 × 5 × 5. Divide by 2: 25 = 5².
  • Medium: 242 = 2 × 11 × 11. Divide by 2: 121 = 11².
  • Exam-type: 3920 = 2⁴ × 5 × 7². Divide by 5: 784 = 28².

Rule 3: Count the numbers between squares — Between n² and (n + 1)² there are 2n numbers. To count squares in a range, find the first and last roots.

  • Easy: Between 7² and 8²: 2 × 7 = 14 numbers.
  • Medium: Between 30² and 31²: 2 × 30 = 60 numbers.
  • Exam-type: Squares from 300 to 700: 17² = 289 is too small, 18² = 324 to 26² = 676 fit, 27² = 729 is too big. So 26 − 18 + 1 = 9 squares.
Remember

Only perfect squares have an odd number of factors. In the locker puzzle, exactly the square-numbered lockers stay open.

Common mistake

Multiplying by the whole number instead of only the single factors: for 252, multiplying by 252 is not the smallest; multiply only by 7.

Shortcut

For "smallest square divisible by several numbers", first find the LCM, then fix its unpaired prime factors. Example: LCM(6, 8) = 24 = 2³ × 3, so multiply by 2 × 3 = 6: 144. Memory clue: "Lonely factor decides": single factor → multiply it in (to make a pair) or divide it out (to remove it). Choose by what the question asks.

Spot the difference

PointMultiply methodDivide method
Single factor ismultiplied in to make a pairdivided out
Example98 × 2 = 1963920 ÷ 5 = 784
The number becomesbiggersmaller

Practice questions

Easy
  1. 1Is 484 a perfect square? (484 = 2² × 11²)
  2. 2How many numbers lie between 12² and 13²?
  3. 3By what should 32 be multiplied to get a perfect square?
Medium
  1. 4By what smallest number should 450 be multiplied to get a perfect square? (450 = 2 × 3² × 5²)
  2. 5By what smallest number should 1452 be divided to get a perfect square? (1452 = 2² × 3 × 11²)
  3. 6How many perfect squares are there between 100 and 400?
Challenging
  1. 7Find the smallest square number divisible by 5, 12 and 14.
  2. 8In the locker puzzle with 400 lockers, how many remain open, and which is the last open locker?
Show answers (try them yourself first)
  1. 1Yes: 484 = 22 × 22
  2. 224
  3. 32 (32 × 2 = 64 = 8²)
  4. 42 (450 × 2 = 900 = 30²)
  5. 53 (1452 ÷ 3 = 484 = 22²)
  6. 69 (11² = 121 up to 19² = 361)
  7. 744100 (LCM 420 = 2² × 3 × 5 × 7, multiply by 105; 210²)
  8. 820 lockers stay open; the last open locker is number 400 (20²)

Cube numbers and their patterns

From daily life

A shopkeeper packs soap boxes in a big carton, 6 boxes along the length, 6 along the width and 6 in height. The carton holds 6 × 6 × 6 = 216 boxes.

Simple definition

The cube of a number is the number multiplied by itself three times: n × n × n = n³. Numbers like 1, 8, 27, 64 are perfect cubes because n³ small cubes make a big cube of edge n. The cube of a negative number is negative: (−3)³ = −27.

layer 1: 9+layer 2: 9+layer 3: 93 × 9 = 27 = 3³

In the figure: A 3 × 3 × 3 cube has 3 layers of 9 small cubes each: 27 in all.

Rule 1: Look at the last digit — The last digit of n³ depends only on the last digit of n. Digits 0, 1, 4, 5, 6, 9 stay the same; 2 and 8 swap; 3 and 7 swap.

  • Easy: 13³ ends in 7, because 3³ = 27.
  • Medium: A cube ending in 8 comes from a number ending in 2 (12³ = 1728).
  • Exam-type: Is 4328 a cube? It ends in 8, so the root must end in 2. But 12³ = 1728 is too small and 22³ is over 10000, so 4328 is NOT a cube.

Rule 2: Cubes and odd numbers — Write the odd numbers in rows of 1, 2, 3, … numbers. The row with n numbers adds up to n³. Row n starts at n × n − n + 1.

  • Easy: 3 + 5 = 8 = 2³.
  • Medium: 13 + 15 + 17 + 19 = 64 = 4³.
  • Exam-type: The 7th row starts at 7 × 7 − 7 + 1 = 43: 43 + 45 + … + 55 = 343 = 7³.

Rule 3: Successive differences — Take differences of cubes again and again: after three levels every difference is 6.

  • Easy: 1, 8, 27, 64 → level 1: 7, 19, 37.
  • Medium: Level 1: 7, 19, 37, 61 → level 2: 12, 18, 24 → level 3: 6, 6.
  • Exam-type: 6³ = 216 and the next level-1 difference is 127, so 7³ = 216 + 127 = 343.
Remember

A negative number has a negative cube, and the cube of an odd number is odd. Taxicab number: 1729 = 1³ + 12³ = 9³ + 10³ is the smallest number that is a sum of two cubes in two ways.

Common mistake

Writing 4³ = 4 × 3 = 12. The cube means 4 × 4 × 4 = 64.

Shortcut

To cube a number quickly, square it first and multiply by the number again: 25³ = 625 × 25 = 15625. Memory clue: Cube-end partners: "2 and 8, 3 and 7 swap (they add to 10); 0, 1, 4, 5, 6, 9 stay the same".

Spot the difference

PointSquaresCubes
Possible last digits0, 1, 4, 5, 6, 9any digit 0 to 9
Negative numbersquare is positivecube is negative
Built from odd numbersfirst n odd numbersthe n numbers in row n

Practice questions

Easy
  1. 1Find 7³.
  2. 2How many 1 cm cubes make a cube of edge 8 cm?
  3. 3Find (−4)³.
Medium
  1. 4What is the units digit of 28³?
  2. 5A cube ends in 7. What can the last digit of the number be?
  3. 6The 9th row of the odd-number pattern starts at 73. Write its sum.
Challenging
  1. 7Can 3024 be a perfect cube? Explain with the last digit.
  2. 8The difference between 8³ and 9³ is 217. Find 9³.
Show answers (try them yourself first)
  1. 1343
  2. 2512
  3. 3−64
  4. 42 (8³ = 512)
  5. 53 (since 3³ = 27)
  6. 6729 (9³): the 9 numbers 73 + 75 + … + 89
  7. 7No: it ends in 4, so the root would end in 4. But 14³ = 2744 and 24³ = 13824, and 3024 is between them
  8. 89³ = 512 + 217 = 729

Cube roots and history of roots

From daily life

A cubical water tank holds 8000 litres, which is 8 m³ (1 m³ = 1000 litres). Its edge is the number whose cube is 8: it is 2 m.

Simple definition

If y = x³, then x is the cube root of y, written x = ∛y. A number has only one cube root, and ∛(−n) = −∛n. Example: ∛27 = 3 and ∛(−27) = −3.

222333216 = (2×2×2) × (3×3×3)one from each triplet: 2 × 3 = 6
Live figure

Cube roots and history of roots

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Rule 1: Group the prime factors in threes — Write the prime factors, make triplets of equal factors and take ONE factor from each triplet. Multiply them.

  • Easy: 216 = (2 × 2 × 2) × (3 × 3 × 3), so ∛216 = 2 × 3 = 6.
  • Medium: 8000 = 2⁶ × 5³, so ∛8000 = 2² × 5 = 20.
  • Exam-type: 74088 = 2³ × 3³ × 7³, so ∛74088 = 2 × 3 × 7 = 42.

Rule 2: Guess with the last digit — For a perfect cube, the last digit gives the last digit of the root, and the size gives the tens digit.

  • Easy: ∛9261: ends in 1, so the root ends in 1; 20³ = 8000 < 9261 < 27000 = 30³, so the root is 21.
  • Medium: ∛13824: ends in 4, so the root ends in 4; it is between 20 and 30, so 24.
  • Exam-type: ∛29791: ends in 1, so the root ends in 1; 30³ = 27000 < 29791 < 64000 = 40³, so the root is 31.

Rule 3: Multiply or divide to make a cube — Find the prime factors, complete every incomplete triplet by multiplying, or remove the leftover factors by dividing.

  • Easy: 54 = 2 × 3³. Multiply by 2 × 2 = 4: 216 = 6³.
  • Medium: 756 = 2² × 3³ × 7. Multiply by 2 × 7 × 7 = 98: 74088 = 42³.
  • Exam-type: 4000 = 2⁵ × 5³. Divide by 2 × 2 = 4: 1000 = 10³.
Remember

History: the Babylonians kept tables of squares and cubes on clay tablets about 1700 BCE. In Sanskrit varga = square, ghana = cube and mula = root, so varga-mula is square root and ghana-mula is cube root. Aryabhata (499 CE) described varga as a square figure with four equal sides.

Common mistake

Dividing by 3: ∛27 = 3, not 9. Check by cubing: 9³ = 729 which is far too big. Also, a cube root uses triplets, not pairs.

Shortcut

Always cube your answer to check it. Remember the root of a cube of a negative number is negative: ∛(−1000) = −10. Memory clue: Square root: "pairs, pick one". Cube root: "THREES, pick one" (Marathi: तिघड्यातून एक). The word "mula" (root) comes from the plant root: the base from which the number grew.

Spot the difference

PointSquare rootCube root
Group the factors inpairstriplets
A positive number hastwo roots (+ and −)one cube root
Example√64 = ±8∛64 = 4

Practice questions

Easy
  1. 1Find ∛343.
  2. 2Find ∛(−64).
  3. 3A cubical box has volume 729 cm³. What is its edge?
Medium
  1. 4Find ∛4096 using prime factors (4096 = 2¹²).
  2. 5Find ∛17576 using the last digit.
  3. 6By what smallest number should 100 be multiplied to get a perfect cube?
Challenging
  1. 7By what smallest number should 3750 be divided to get a perfect cube? (3750 = 2 × 3 × 5⁴)
  2. 8Find ∛(27 × 1000) + ∛(−8).
Show answers (try them yourself first)
  1. 17
  2. 2−4
  3. 39 cm
  4. 416
  5. 526
  6. 610 (100 = 2² × 5², multiply by 2 × 5)
  7. 730 (3750 ÷ 30 = 125 = 5³)
  8. 828 (30 − 2)