CBSE Class 8 Maths Revision Notes Chapter 1: A Square and A Cube

Square numbers are obtained by multiplying a number by itself, while cube numbers are obtained by multiplying it by itself three times. Their patterns, factors and roots help solve problems related to numbers, areas, volumes and estimation.

The chapter begins with a locker puzzle in which only lockers numbered by perfect squares remain open. This happens because perfect squares are the only natural numbers that have an odd number of factors.

These CBSE Class 8 Maths Revision Notes Chapter 1 cover square numbers, perfect squares, square roots, cube numbers, perfect cubes, cube roots, number patterns, prime factorisation and estimation. The chapter also introduces the Hardy–Ramanujan number 1729 and explains the historical Indian terms used for squares, cubes and roots.

Key Takeaways

• Perfect square: A number written in the form n², where n is a natural number.
• Perfect cube: A number written in the form n³.
• Square root: A number which gives the original number when multiplied by itself.
• Cube root: A number which gives the original number when multiplied by itself three times.

Access Class 8 Maths Chapter 1 A Square and A Cube Notes in 30 Minutes

Spend the first 10 minutes revising square numbers, factor pairs and properties of perfect squares. Use the next 10 minutes for square roots, prime factorisation and estimation.

Use the final 10 minutes to revise cubes, cube roots, perfect-cube patterns and the Hardy–Ramanujan number.

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Square Numbers in Class 8 Maths Chapter 1 Notes

A square number is obtained by multiplying a number by itself. If the number is n, its square is written as n².

n² = n × n

Examples:

1² = 1 × 1 = 1
2² = 2 × 2 = 4
3² = 3 × 3 = 9
4² = 4 × 4 = 16
5² = 5 × 5 = 25

The word square is connected with geometry. A square having a side of n units has an area of n² square units.

Squares can also be calculated for fractions and decimals.

(3/5)² = 3/5 × 3/5 = 9/25

(2.5)² = 2.5 × 2.5 = 6.25

What Is a Perfect Square?

The square of a natural number is called a perfect square. The first few perfect squares are:

1, 4, 9, 16, 25, 36, 49, 64, 81 and 100

Why Do Perfect Squares Have an Odd Number of Factors?

Factors of a number generally occur in pairs. For example:

6 = 1 × 6
6 = 2 × 3

Therefore, the factors of 6 are 1, 2, 3 and 6. It has four factors, which is an even number.

In a perfect square, one factor pair contains the same number twice. For example:

36 = 6 × 6

The complete factors of 36 are:

1, 2, 3, 4, 6, 9, 12, 18 and 36

There are nine factors. The factor 6 is paired with itself, so one factor remains unpaired. This is why every perfect square has an odd number of factors.

Locker Puzzle and Perfect Squares

A locker is toggled once for every factor of its locker number. A locker remains open only when it is toggled an odd number of times.

Only perfect squares have an odd number of factors. Therefore, the lockers that remain open are:

1, 4, 9, 16, 25, 36, 49, 64, 81 and 100

The lockers touched exactly twice are numbered by prime numbers because a prime number has only two factors, 1 and the number itself. The first five such lockers are:

2, 3, 5, 7 and 11

Properties of Perfect Squares

Perfect squares follow several useful patterns. These patterns help identify numbers that cannot be perfect squares.

Possible Units Digits of Perfect Squares

A perfect square can end only in:

0, 1, 4, 5, 6 or 9

A perfect square cannot end in:

2, 3, 7 or 8

For example, 327 cannot be a perfect square because its units digit is 7.

However, a number ending in 0, 1, 4, 5, 6 or 9 is not always a perfect square. For example, 16 is a perfect square, but 26 is not, even though both numbers end in 6.

Units Digit of a Number and Its Square

Examples:

34² = 1156
46² = 2116
74² = 5476

Numbers ending in 4 or 6 have squares ending in 6.

Zeros at the End of a Perfect Square

If a number ends with n zeros, its square ends with 2n zeros.

10² = 100

One zero in 10 becomes two zeros in its square.

100² = 10,000

Two zeros in 100 become four zeros in its square.

1,000² = 1,000,000

Three zeros in 1,000 become six zeros in its square.

Therefore, a perfect square can have only an even number of zeros at the end.

Even and Odd Square Numbers

The square of an even number is even.

6² = 36

The square of an odd number is odd.

7² = 49

Perfect Squares and Consecutive Odd Numbers

The difference between two consecutive perfect squares is always an odd number.

2² − 1² = 4 − 1 = 3

3² − 2² = 9 − 4 = 5

4² − 3² = 16 − 9 = 7

5² − 4² = 25 − 16 = 9

This gives the following pattern:

1 = 1²

1 + 3 = 4 = 2²

1 + 3 + 5 = 9 = 3²

1 + 3 + 5 + 7 = 16 = 4²

1 + 3 + 5 + 7 + 9 = 25 = 5²

Therefore, the sum of the first n odd natural numbers is n².

1 + 3 + 5 + ... + (2n − 1) = n²

Finding the Next Square

Suppose 35² = 1225 and we need to find 36².

The nth odd number is:

2n − 1

The 36th odd number is:

2 × 36 − 1 = 71

Therefore:

36² = 35² + 71

36² = 1225 + 71

36² = 1296

Checking a Perfect Square by Subtracting Odd Numbers

A natural number is a perfect square if repeated subtraction of consecutive odd numbers beginning with 1 ends exactly at 0.

For 25:

25 − 1 = 24
24 − 3 = 21
21 − 5 = 16
16 − 7 = 9
9 − 9 = 0

Since 25 reaches 0 after five subtractions:

25 = 5²

Now consider 38:

38 − 1 = 37
37 − 3 = 34
34 − 5 = 29
29 − 7 = 22
22 − 9 = 13
13 − 11 = 2
2 − 13 = −11

The result crosses 0 instead of reaching exactly 0. Therefore, 38 is not a perfect square.

Numbers between Consecutive Perfect Squares

The number of natural numbers between n² and (n + 1)² is 2n.

For example:

4² = 16

5² = 25

The numbers between 16 and 25 are:

17, 18, 19, 20, 21, 22, 23 and 24

There are 8 numbers.

2 × 4 = 8

Similarly, the number of natural numbers between 16² and 17² is:

2 × 16 = 32

The number of natural numbers between 99² and 100² is:

2 × 99 = 198

Square Roots in A Square and A Cube Notes

The square root of a number is a value which, when multiplied by itself, gives the original number.

If x² = y, then x is a square root of y.

For example:

7² = 49

Therefore:

√49 = 7

Both 7 and −7 give 49 when squared.

7² = 49

(−7)² = 49

Thus, the integer square roots of 49 are +7 and −7. In this chapter, we mainly use the positive square root.

Common Square Roots

Finding Square Roots by Prime Factorisation

A number is a perfect square if every prime factor occurs an even number of times. To find its square root, form pairs of equal prime factors and take one factor from each pair.

Example: Find √324

Prime factorisation:

324 = 2 × 2 × 3 × 3 × 3 × 3

Group equal factors in pairs:

324 = (2 × 2) × (3 × 3) × (3 × 3)

Take one factor from each pair:

√324 = 2 × 3 × 3

√324 = 18

Therefore:

324 = 18²

Is 156 a Perfect Square?

Prime factorisation:

156 = 2 × 2 × 3 × 13

The factors 3 and 13 do not have equal partners. Therefore, all prime factors cannot be arranged in pairs.

Hence, 156 is not a perfect square.

Making a Number a Perfect Square

A number can be multiplied by a suitable factor so that every prime factor appears an even number of times.

Consider 9408:

9408 = 2⁶ × 3 × 7²

The factor 3 appears only once. Multiply the number by 3:

9408 × 3 = 2⁶ × 3² × 7²

Now all powers are even.

√(9408 × 3) = 2³ × 3 × 7

√28,224 = 168

Therefore, the smallest number by which 9408 must be multiplied is 3.

Estimating Square Roots

A square root can be estimated by placing the given number between two nearby perfect squares.

Example: Estimate √250

15² = 225

16² = 256

Therefore:

15 < √250 < 16

Since 250 is closer to 256 than to 225, √250 is approximately 16. Its actual value is slightly less than 16.

Example: Largest Square from an Area of 125 cm²

11² = 121

12² = 144

A square with a side of 12 cm requires 144 cm², which is more than 125 cm². Therefore, the largest square with an integer side length has a side of 11 cm.

Estimating the Square Root of a Large Perfect Square

Consider √1936.

40² = 1600

50² = 2500

Therefore:

40 < √1936 < 50

The number 1936 ends in 6, so its square root must end in 4 or 6. The possible answers are 44 and 46.

Now:

45² = 2025

Since 1936 is less than 2025, its square root is less than 45.

Therefore:

√1936 = 44

Cube Numbers in Class 8 Maths Chapter 1 Revision Notes

A cube number is obtained by multiplying a number by itself three times.

n³ = n × n × n

Examples:

1³ = 1 × 1 × 1 = 1

2³ = 2 × 2 × 2 = 8

3³ = 3 × 3 × 3 = 27

4³ = 4 × 4 × 4 = 64

The term cube is connected with geometry. A cube with an edge of n units contains n³ unit cubes.

For example, a cube with an edge of 4 units contains:

4 × 4 × 4 = 64 unit cubes

What Is a Perfect Cube?

The cube of a natural number is called a perfect cube.

The first few perfect cubes are:

1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000

Cubes can also be calculated for fractions, decimals and negative numbers.

(4/6)³ = 4/6 × 4/6 × 4/6 = 64/216

(−6)³ = −6 × −6 × −6 = −216

The cube of a negative number is negative because three negative factors are multiplied.

Units Digits of Perfect Cubes

A cube may end in any digit from 0 to 9. However, the units digit of a cube follows a fixed pattern.

Examples:

12³ = 1728

The number 12 ends in 2, so its cube ends in 8.

17³ = 4913

The number 17 ends in 7, so its cube ends in 3.

Zeros at the End of a Perfect Cube

If a number ends with n zeros, its cube ends with 3n zeros.

10³ = 1000

One zero becomes three zeros.

100³ = 1,000,000

Two zeros become six zeros.

Therefore, a perfect cube cannot end with exactly two zeros because the number of trailing zeros in a perfect cube must be a multiple of three.

Perfect Cubes and Consecutive Odd Numbers

Perfect cubes can also be written as sums of consecutive odd numbers.

1 = 1 = 1³

3 + 5 = 8 = 2³

7 + 9 + 11 = 27 = 3³

13 + 15 + 17 + 19 = 64 = 4³

21 + 23 + 25 + 27 + 29 = 125 = 5³

Each n³ can be expressed as the sum of n consecutive odd numbers.

Cube Roots

The cube root of a number is a value which, when multiplied by itself three times, gives the original number.

If x³ = y, then x = ∛y.

Examples:

∛8 = 2

∛27 = 3

∛64 = 4

∛125 = 5

∛1000 = 10

Unlike square roots, the cube root of a negative number is also negative.

∛−216 = −6

This is because:

−6 × −6 × −6 = −216

Finding Cube Roots by Prime Factorisation

A number is a perfect cube when every prime factor occurs in a group of three. To find the cube root, form triplets of equal prime factors and take one factor from each triplet.

Example: Find ∛3375

Prime factorisation:

3375 = 3 × 3 × 3 × 5 × 5 × 5

Group the equal factors:

3375 = (3 × 3 × 3) × (5 × 5 × 5)

Take one factor from each triplet:

∛3375 = 3 × 5

∛3375 = 15

Therefore:

3375 = 15³

Is 500 a Perfect Cube?

Prime factorisation:

500 = 2 × 2 × 5 × 5 × 5

The factors 5 form one triplet, but the two factors of 2 do not form a complete triplet. Therefore, 500 is not a perfect cube.

Making a Number a Perfect Cube

To make a number a perfect cube, every prime factor must occur in a multiple of three.

Consider 1323:

1323 = 3³ × 7²

The factor 7 occurs twice. Multiplying by another 7 forms a complete triplet.

1323 × 7 = 3³ × 7³

Therefore, the smallest number by which 1323 must be multiplied is 7.

Hardy–Ramanujan Number 1729

The number 1729 is called the Hardy–Ramanujan number. It is the smallest number that can be expressed as the sum of two positive cubes in two different ways.

1729 = 1³ + 12³

1729 = 9³ + 10³

Check:

1³ + 12³ = 1 + 1728 = 1729

9³ + 10³ = 729 + 1000 = 1729

Numbers that can be expressed as the sum of two positive cubes in two different ways are called taxicab numbers.

Squares and Cubes in Indian Mathematical History

Ancient Indian mathematics used special terms for squares, cubes and roots.

• Varga: Square figure, square area or square power
• Ghana: Cube or cube power
• Varga-mula: Square root
• Ghana-mula: Cube root
• Mula: Root, base, cause or origin

Aryabhata used the word varga for a square figure and for the product of two equal quantities. The mathematical term root developed from the idea of a root as the base or origin of something.

Square and Cube Comparison

Class 8 Maths Chapter 1 Summary Notes

Important Terms from A Square and A Cube

Square number: A number obtained by multiplying a number by itself.

Perfect square: The square of a natural number.

Square root: A number which gives the original number when multiplied by itself.

Factor pair: Two factors whose product gives a number.

Cube number: A number obtained by multiplying a number by itself three times.

Perfect cube: The cube of a natural number.

Cube root: A number which gives the original number when multiplied by itself three times.

Prime factorisation: Writing a number as a product of prime factors.

Estimation: Finding a value close to the exact answer.

Taxicab number: A number that can be expressed as the sum of two positive cubes in two different ways.

Hardy–Ramanujan number: The number 1729.

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