Cubes And Cube Roots

Chapter 10: Cubes and Cube Roots

This Class 8 Maths chapter teaches students how to find cubes and cube roots of numbers using prime factorization and estimation methods. The notes include properties of cubes, patterns, and relations with perfect numbers. Students practice identifying perfect cubes and calculating cube roots through step-by-step examples. Shortcut methods and tables help in mental math calculations. These Class 8 Maths Notes make the topic easy for revision and provide NCERT-based exercises with solutions, ensuring full marks in exam questions related to cubes and cube roots.

Cubes and Cube Roots Theory - Complete Guide

Cube Number

A cube number (or a cube) is a number you can write as a product of three equal factors of natural numbers.

Formula: k = a × a × a = a³

(k and a stand for integers)

A cube number results by multiplying an integer by itself three times.

Formula: a × a × a = a³ = k

How to Cube a Nummber

To cube a number, just use it in a multiplication 3 times.

1. What is 3 Cubed?

Sol. 3 Cubed = 3 × 3 × 3 = 27

Note: We write down "3 Cubed" as 3³ (the little "3" means the number appears three times in multiplying)

Some More Cubes

Expression	Exponential Form	Calculation	Result
4 cubed	4³	4 × 4 × 4	64
5 cubed	5³	5 × 5 × 5	125
6 cubed	6³	6 × 6 × 6	216

Properties of Cube of a Number

• The cube of every even number is even.
• The cube of every odd number is odd.

Cubes of Negative Integers

The cube of a negative integer is always negative.

Example:

(-1)³ = (-1) × (-1) × (-1) = -1

(-2)³ = (-2) × (-2) × (-2) = -8

(-3)³ = (-3) × (-3) × (-3) = -27

etc.

Cube of a Rational Number

For a rational number a/b, the cube is given by:

(a/b)³ = a³/b³

For example:

(i) (2/3)³ = 2³/3³ = 8/27

(ii) (-3/5)³ = (-3)³/5³ = -27/125

Cube Root

The cube roots of a number x are the numbers y which satisfy the equation:

y³ = x

A cube root of a number, denoted ∛x or x^1/3, is a number a such that a³ = x. All real numbers (except zero) have exactly one real cube root and a pair of complex roots.

For example: The real cube root of 8 is 2, because 2³ = 8.

Understanding Cube Root

A cube root goes the other direction:

3 cubed is 27, so the cube root of 27 is 3

The cube root of a number is the special value that when cubed gives the original number.

The cube root of 27 is 3, because when 3 is cubed you get 27.

1. What is the Cube root of 125?

Sol. Well, we just happen to know that 125 = 5 × 5 × 5 (if you use 5 three times in a multiplication you will get 125)

... so the answer is 5

The Cube Root Symbol

This is the special symbol that means "cube root": ∛

It is the "radical" symbol (used for square roots) with a little three to mean cube root.

You can use it like this: ∛27 = 3 (you would say "the cube root of 27 equals 3")

You Can Also Cube Negative Numbers

Have a look at this:

If you cube 5 you get 125: 5 × 5 × 5 = 125

If you cube -5 you get -125: (-5) × (-5) × (-5) = -125

So the cube root of -125 is -5

Perfect Cubes

The Perfect Cubes are the cubes of the whole numbers:

Number	1	2	3	4	5	6	7	8	9	10
Perfect Cubes	1	8	27	64	125	216	343	512	729	1000

Number	11	12	13	14	15
Perfect Cubes	1331	1728	2197	2744	3375

Properties of Cube Root of a Number

1. Cube Root of a Negative Perfect Cube

Let a be a positive integer. Then, (-a) is a negative integer. We know that:

(-a)³ = (-a) × (-a) × (-a) = -a³

∴ ∛(-a³) = -a

Thus, cube root of (-a³) is -(cube root of a³)

Examples:

∛(-8) = ∛[(-2)³] = -2

∛(-27) = ∛[(-3)³] = -3

∛(-64) = ∛[(-4)³] = -4

∛(-125) = ∛[(-5)³] = -5

∴ ∛(-1000) = ∛[(-10)³] = -10

2. Cube Root of a Product of Integers

∛(ab) = ∛a × ∛b

Example:

∛(8 × 27) = ∛8 × ∛27

∛216 = 2 × 3

6 = 6

3. Cube Root of a Rational Number

∛(a/b) = ∛a / ∛b

Example:

∛(8/27) = ∛8 / ∛27 = 2/3

4. Important Note

The cube root operation is not associative or distributive with addition or subtraction.

5. Associative and Distributive Properties

The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers.

Example: ∛(-27) = -3

6. Relationship Between Unit's Digit of Cube and Its Cube Root

Unit's digits of cube	Unit's digit of cube root
0	0
1	1
2	8
3	7
4	4
5	5
6	6
7	3
8	2
9	9

How to Find Cube of a Given Number

Short-cut Method for Finding the Cube of a Two Digit Number

We have: (10a + b)³ = 1000a³ + 300a²b + 30ab² + b³

For finding the cube of a two-digit number with the tens digit = a and the unit's digit = b, we make four columns, headed by a³, 3a²b, 3ab², and b³

1. Find the cubes of 35 by the short-cut method.

Sol. Here a = 3 and b = 5

a³	3a²b	3ab²	b³
27	135	225	125
27	13 | 5	22 | 5	12 | 5
27 + 13 = 40, 40 + 5 = 45, 45 + 22 = 67, 67 + 5 = 72, 72 + 12 = 84, 84 + 5 = 89

∴ 35³ = 42875

2. Find the smallest whole number by which each of the following numbers must be divided to obtain a perfect cube:

(a) 81 (b) 128 (c) 135 (d) 192 (e) 704

Sol. (a) 81 = 3 × 3 × 3 × 3

The prime factor 3 does not appear in a group of three. So if we divide 81 by 3, then the prime factorisation of the quotient will not contain 3.

81 ÷ 3 = 3 × 3 × 3

Hence the smallest whole number by which 81 should be divided to make it perfect cube is 3.

(b) 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

The prime factor 2 does not appear in a group of three. So if we divide 128 by 2, then the prime factorisation of the quotient will not contain 2.

128 ÷ 2 = 2 × 2 × 2 × 2 × 2 × 2

Hence the smallest whole number by which 128 should be divided to make it perfect cube is 2.

(c) 135 = 3 × 3 × 3 × 5

In the factorization 5 appears only one time. So if we divide 135 by 5, then the prime factorization of the quotient will not contain 5.

135 ÷ 5 = 3 × 3 × 3

Hence the smallest whole number by which 135 should be divided to make it perfect cube is 5.

(d) 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3

In the factorization 3 appears only one time. So, if we divide 192 by 3, then the prime factorization of the quotient will not contain 3.

192 ÷ 3 = 2 × 2 × 2 × 2 × 2 × 2

Hence, the smallest whole numbers by which 192 should be divided to make it perfect cube is 3.

(e) 704 = 2 × 2 × 2 × 2 × 2 × 2 × 11

In the factorization 11 appears only one time. So if we divide 704 by 11, then the prime factorization of the quotient will not contain 11.

704 ÷ 11 = 2 × 2 × 2 × 2 × 2 × 2

Hence the smallest whole number by which 704 should be divided to make it perfect cube is 11.

4. State true or false.

• (a) Cube of any odd number is even
• (b) A perfect cube does not end with two zeros
• (c) If square of a number ends with 5, then its cube ends with 25.
• (d) There is no perfect cube which ends with 8.
• (e) The cube of a two digit number may be a three digit number.
• (f) The cube of a two digit number may have seven or more digits.
• (g) The cube of a single digit number may be a single digit number.

Sol. (a) False (b) True (c) False (d) False

(e) False (f) False (g) True

5. Is 53240 a perfect cube? If not, then by which smallest natural number should 53240 be divided so that the quotient is a perfect cube?

Sol. 53240 = 2 × 2 × 2 × 11 × 11 × 11 × 5

The prime factor 5 does not appear in a group of three. So, 53240 is not a perfect cube. In the factorization 5 appears only one time. If we divide the number by 5, then the prime factorization of the quotient will not contain 5.

Hence the smallest number by which 53240 should be divided to make it a perfect cube is 5.

The perfect cube in that case is = 10648.

6. Observe the following pattern of sums of 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

31 + 33 + 35 + 37 + 39 + 41 = 216 = 6³

Express the following numbers as the sum of odd numbers using the above pattern?

(a) 7³ (b) 8³ (c) 9³

Sol. According to above pattern, we have:

(a) 7³ = 43 + 45 + 47 + 49 + 51 + 53 + 55 = 343

(b) 8³ = 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71 = 512

(c) 9³ = 73 + 75 + 77 + 79 + 81 + 83 + 85 + 87 + 89 = 729

How to Find Cube Root of a Given Number

Cube Root Through a Pattern

First Pattern:

∛1 = 1 (1 = 1³)

∛8 = 2 also 1 + 7 = 8 = 2³

∛27 = 3 also 1 + 7 + 19 = 27 = 3³

∛64 = 4 also 1 + 7 + 19 + 37 = 64 = 4³

∛125 = 5 also 1 + 7 + 19 + 37 + 61 = 125 = 5³

Second Pattern:

∛1 = 1 = 1 - 0 = 1² - 0²

∛8 = 2 = 4 - 2 = 2² - (√2)²

∛27 = 3 = 9 - 6 = 3² - (√6)²

∛64 = 4 = 16 - 12 = 4² - (√12)²

∛125 = 5 = 25 - 20 = 5² - (√20)²

Thus, ∛n³ = n = n² - (√(n² - n))²

Cube Root Using Unit's Digit

⮊ The cube of a number ending in 0, 1, 4, 5, 6 and 9 ends in 0, 1, 4, 5, 6 and 9 respectively.

⮊ The cube of a number ending in 2 ends in 8 and vice-versa.

⮊ Similarly, the cube of a number ending in 3 or 7 ends in 7 or 3 respectively.

Thus, by looking at the unit's digit of a perfect cube number, we can find the unit's digit of the cube-root.

Cube Root by Prime Factorization

To find ∛n, follow the following steps:

(i) Find the prime factorization of 'n'.

(ii) Group the factors in triples such that all three factors in each triple are the same.

(iii) If some prime factors are left ungrouped, the number 'n' is not a perfect cube and the process stops.

(iv) If no factor is left ungrouped, choose one factor from each group and take their product.

(v) The product is the required cube root of 'n'.

Cube Root by Hand

STEPS:

1. Write down the number of which you wish to calculate the cube root, separating the digits into groups of three, starting at the decimal point, from both directions. Draw a cube radical sign over the number, and put a decimal point over the radical directly above the decimal point in the number. For example, let us calculate the cube root of 10, so we separate it as 10. 000 000.

2. Start with the leftmost group of number(s), and find the biggest integer whose cube is less than or equal to it. Write the integer above the radical, and its cube under the first group. Draw a line under that cube, and subtract it from the first group. In our example, 2³ = 8 < 10 < 3³ = 27, so write 2 over the radical, write 8 under the first group, and subtract it from the first group, resulting in 2.

3. Bring down the next group of numbers into the remainder, and draw a vertical line to the left of the resulting number. To the left of the vertical line, write three hundred times the square of the number above the radical, a plus sign, thirty times the number above the radical, a multiplication sign, an underscore character, another plus sign, another underscore character, the exponent 2, an equals sign, and some blank space for the answer. For our example, bring down the three 0's. 300 times square of 2 is 1200, 30 times 2 is 60, so write "1200 + 60 × _ + _² = (blank space)" to the left of the vertical line.

4. Find the biggest integer N that would fit into both underscore places, and give a number in the blank space such that integer N times the number is less than the current remainder. Put the integer N above the radical and into both underscore places, calculate the number to the right of the equal sign, multiply this number by N, write the product under the current remainder, draw a line under that, and subtract to obtain the new remainder. For our example, the integer is 1, 1200 + 60 × 1 + 1² = 1261, and 1 × 1261 = 1261, which subtracted from 2000 is 739. If the current answer above the radical has the desired accuracy, stop. Otherwise, proceed to the next step.

5. Repeat the previous two steps to find the next digit in the cube root.

The result above the radical is the cube root, accurate to three significant figures. In our example, the cube root of 10 is 2.15. Verify that by calculating 2.15³ = 9.94, which approximates 10. If you need greater accuracy, simply continue the process.

Questions with Detailed Solutions

1. Find the cube root of 74088 by using unit's digit.

Sol. The unit's digit of 74088 is 8

∴ The unit's digit of its cube root is 2

i.e. ∛74088 = __2

After striking out the last three digits from the right, we are left with 74.

Now 4³ = 64 < 74 < 5³ = 125

∴ The ten's digit of the cube root is 4.

Hence ∛74088 = 42

2. Find the cube root of 17576 through estimation.

Sol. The given number is 17576.

Form groups of three starting from the rightmost digit of 17576.

17 | 576. In this case one group i.e., 576 has three digits whereas 17 has only two digits.

Take 576.

The digit 6 is at its one's place.

Take the one's place of the required cube root as 6.

Take the other group, i.e., 17.

Cube of 2 is 8 and cube of 3 is 27. 17 lies between 8 and 27.

The smaller number among 2 and 3 is 2.

The one's place of 2 is 2 itself. Take 2 as ten's place of the cube root of 17576.

Thus, ∛17576 = 26.

3. Which of the following numbers are not perfect cubes?

(a) 216 (b) 128 (c) 1000 (d) 100 (e) 46656

Sol:

(a) 216 = 2 × 2 × 2 × 3 × 3 × 3

= 2³ × 3³

= (2 × 3)³

= 6³

Which is a perfect cube.

(b) 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

= 2³ × 2³ × 2

∴ 2 does not appear in a group of three.

Hence, 128 is not a perfect cube.

(c) 1000 = 2 × 2 × 2 × 5 × 5 × 5

= 2³ × 5³

= (2 × 5)³

= 10³

which is a perfect cube.

(d) 100 = 2 × 2 × 5 × 5

Prime factor of 100 is 2 × 2 × 5 × 5

So, 100 is not a perfect cube.

(e) 46656 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

= 2³ × 2³ × 3³ × 3³

= (2 × 2 × 3 × 3)³

= 36³

which is a perfect cube.

4. You are told that 1,331 is a perfect cube. Can you guess without factorization what its cube root is? Similarly, guess the cube roots of 4913 and 12167.

Sol. The given number is 1331

• Step 1: Form groups of three starting from the rightmost digit of 1331. In this case one group 331 has three digits whereas 1 has only one digit.
• Step 2: Take 331. The digit 1 is at its one's place. We take the one's place of the required cube root as 1.
• Step 3: Take the other group is 1. Cube of 1 is 1 and cube of 2 is 8. 1 lies between 0 and 8. The smaller number between 1 and 2 is 1. The one's place of 1 is 1 itself. Take 1 as ten's place of the cube root of 1331.

Thus, ∛1331 = 11

Similarly for 4913, we have:

• Step 1: Form groups of three starting from the rightmost digit of 4913
• Step 2: Take 913. The digit 3 is at its one's place we take the one's place of the required cube root as (3 × 3 × 3 = 27) 7
• Step 3: Take the other group is 4. Cube of 1 is 1 and cube of 2 is 8. 4 lies between 1 and 8. The smaller number between 1 and 2 is 1. Take 1 as ten's place of the cube root of 4913.

Thus, ∛4913 = 17

Similarly for 12167, we have:

• Step 1: Form groups of three starting from the rightmost digit of 12167
• Step 2: 7 × 7 × 7 = 343 i.e. one's place is 3
• Step 3: 12, 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27

8 < 12 < 27

The smaller number between 2 and 3 is 2.

The one's place of 2 is 2 itself. Take 2 as tens place of the cube root of 12167.

Thus, ∛12167 = 23

5. Three numbers are in the ratio 1 : 2 : 3. The sum of their cubes is 98784. Find the numbers.

Sol. Let the numbers be x, 2x and 3x. Then:

x³ + (2x)³ + (3x)³ = 98784

⇒ x³ + 8x³ + 27x³ = 98784

⇒ 36x³ = 98784

⇒ x³ = 98784/36

⇒ x³ = 2744

⇒ x = ∛2744

⇒ x = 14

⇒ 2x = 2 × 14 = 28

⇒ 3x = 3 × 14 = 42

Hence the numbers are 14, 28, and 42

Detailed Solutions of Additional Problems

6. Find the smallest number by which 8192 must be divided so that quotient is a perfect cube. Also, find the cube root of the quotient so obtained.

Sol. Resolving 8192 into factors we get:

8192 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

For a number to be perfect cube each of its prime factors must occur thrice. Therefore the smallest number by which the given number must be divided in order that the quotient is a perfect cube is 2.

Also the quotient,

8192/2 = 4096 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

So, ∛4096 = ∛(2³ × 2³ × 2³ × 2³)

= 2 × 2 × 2 × 2

∴ ∛4096 = 16

7. Find the cube root of 13824/6859

Sol. ∛(13824/6859) = ∛13824 / ∛6859

∴ ∛(13824/6859) = 24/19

8. Find the cube root of

(a) -5823

(b) -216 × 1728

(a) We have, -5823 = -(3 × 3 × 3 × 6 × 6 × 6)

= -[(3 × 6)³]

So, ∛(-5823) = ∛[-(18)³]

= -18

= -18

(b) -216 × 1728 = -(216 × 1728)

Now resolving 216 and 1728 into prime factors.

∴ 216 = 2 × 2 × 2 × 3 × 3 × 3

= (2 × 3)³

= 6³

and, 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

= (2 × 2 × 3)³

= 12³

Hence, ∛(-216 × 1728) = ∛[-(6³ × 12³)]

= -(6 × 12)

= -72

9. Find the cube root of 42875.

Sol. The factors of 42875 are 5, 5, 5, 7, 7, 7

∴ 42875 = 5 × 5 × 5 × 7 × 7 × 7 = 5³ × 7³ = (5 × 7)³

∴ ∛42875 = 5 × 7

= 35

10. Find the least number by which 432 must be multiplied or divided to make it a perfect cube.

Sol. The factors of 432 are 3, 3, 3, 4, 4 = (3)³ × (4)²

We note that the factor 4 appears only 2 times.

1. If we multiply 432 by 4 we get 432 × 4 = (3)³ × (4)³ = (3 × 4)³, so when the smallest number 4 multiplies 432, it gives us the cube number (= 1728 = 12³)

2. If we divide 432 by 16 we get 432/16 = 3³, so when the smallest number 16 divides 432, it gives us the cube number (= 27 = 3³).