CBSE Class 8 Mathematics Cubes And Cube Roots Worksheet Set 01

Very Short Answer type Questions

Question. Find the value of 3√343 – 3√–216.
Answer : 13

Question. Find the cube root of –1728.
Answer : –12

Question. Simplify 2√216/2√8 x 3√512/3√27
Answer : 8

Question. Simplify : 3√216 x 3√1728.
Answer : 72

Question. How much is 3√–0.729 ?
Answer : –0.9

Question. (0.3)³ = _________________.
Answer : 0.027

Question. If volume of a cube is 216 cm³. What is the length of side of cube.
Answer : 6cm

Question. Find the smallest number by which (2 × 2 × 3 × 3 × 3) is to be multiplied so that resultant number is a perfect cube.
Answer : 2

Question. What is the value of (–0.2)³?
Answer : –0.008

Question. Find the value of 3√125 x 343.
Answer : 35

Question. Find the value of (3√2/5)³ + (3√3/5)³
Answer : 1

Question. Find 3√x if x = 1.331
Answer : 1.1

Question. How many thousands will be there is 29 × 2³ × 5³?
Answer : 29

Question. What is the next number in the series 64, 125, 216, ____?
Answer : 343

Question. Three cubes of sides 3cm, 4cm and 5 cm respectively are melted to form a new cube.
What is the side of new cube?
Answer : 6cm

Question. Find the next number is the series 3(√8) , 3(√27) , 3(√64) , _________
Answer : 3√125 = 5

<1M>

1.Which of the following is a perfect cube.
(A) 100
(B) 68600
(C) 343000
(D) 400

2.What is smallest number by which 392 must be multiplied so that the product is a perfect cube.
(A) 7
(B) 2
(C) 5
(D) 3

3.Find the cube of 18.

4.Find the one's digit of the cube of each of the following
(a) 1024
(b) 77

5.Find the cube root of 512.

6.Evaluate (0.8)3

7.Cube of 6 is
(A) 36
(B) 216
(C) 96
(D) 26

8.23 can be written as sum of consecutive odd numbers as
(A) 3 + 5
(B) 1 + 3 + 3 +1
(C) 1 + 5 +3
(D) 4 + 3 + 1

9.Prime factors of 216 are:

10.The cube of every even number is
(A) Odd
(B) Even
(C) Neither odd nor even
(D) Both

11.Which one of the following is the cube of an odd number.
(A) 64
(B) 343
(C) 8000
(D) 1728

12.The cube of a number is that number raise to the power.
(A) 6
(B) 2
(C) 3
(D) 1

13.The cube root of a number 'x' is the number whose cube is:
(A) 1
(B) x
(C) 0
(D) None

14.In the prime factorization of any number each factor appears three times, then the number is a:
(A) Perfect cube
(B) Even
(C) Odd
(D) None

15.A perfect cube number does not end with ------------------ zeros.
(A) Two
(B) Three
(C) Six
(D) Nine

16.63 is expressed as the sum of consecutive odd numbers as:
(A) 23+25+27+29+31
(B) 31+33+35+37+39+41
(C) 13+15 +17+19+21+23
(D) 21+23+25+27+29

17.The cube of149 one's digit is:
(A) 2
(B) 9
(C) 3
(D) 1

18.Which of the following is a perfect cube:
(A) 400
(B) 8000
(C) 10000
(D) 100

19.The value of 73 - 63 is:
(A) 127
(B) 13
(C) 243
(D) 143

20.The cube of 53 one's digit is
(A) 4
(B) 6
(C) 9
(D) 27

21.How many times each prime factor appears in its cubes
(A) 1
(B) 2
(C) 3
(D) 4

22.Using the pattern given in book, the value of 123 - 113 is
(A) 397
(B) 227
(C) 427
(D) 87

23.Figures which have 3 dimensions are known as
(A) Plane figures
(B) Solid figures
(C) Two dimension figures
(D) None

24.The smallest number which can be expressed as a sum of two cubes in two different ways is
(A) 1408
(B) 1178
(C) 1729
(D) 1658

25.Cube root of 4913 is
(A) 12
(B) 17
(C) 13
(D) 27

26.Which one of the following is a perfect cube.
(A) 216
(B) 100
(C) 128
(D) 8130

27.The unit digit of the cube of 729 is
(A) 1
(B) 2
(C) 9
(D) 3

28.The volume of a cuboid of sides 5 cm, 3 cm, 2 cm is
(A) 30 cm³
(B) 20 cm³
(C) 40 cm³
(D) 10 cm³

29.The cube of 5022 one's digit is
(A) 8
(B) 9
(C) 4
(D) 6

<2M>

30.Find the smallest number by which 704 must be divided to obtain a perfect cube.

31.Find the cube root of 4096 by primefactorization

32.Find the cube root of 2197 by estimation

33.Show that 189 is not a perfect cube.

34.Find the number whose cube is 9261.

35.The smallest number by which 100 must be multiplied to obtained a perfect cube.
(A) 10
(B) 20
(C) 40
(D) 30

36.The smallest number by which 192 must be divided to obtain a perfect cube is
(A) 2
(B) 3
(C) 4
(D) 1

37.Find the cube of 25

38.Find the cube root of 1.331.

39.Evaluate cube root of 216 × 1728

40.Find the cube root of 5832 by primefactorisation

41.Find the cube root of 5832.

42.Evaluate (i) cube root of 125 × 27= 3× ---------, (ii) cube root of 8 × ----- = 8

<4M>

43.Find the cube root of 91125.

44.The smallest number by which 1188 should be divided to make is a perfect cube is

45.Find the smallest number by which 1600 must be divided so that the quotient is a perfect cube, further find its cube root.

46.Sheetal makes a cuboid of plastics of sides 5 cm, 2 cm, 5cm. How many such cuboids will be need to form a cube?

47.Find the smallest number by which 12500 must be multiplied so that the product is a perfect cube.

48.By prime factorization method, the cube root of 13824 is
(A) 34
(B) 24
(C) 14
(D) 34

<6M>

49.Find the smallest number which when multiplied with 3600 will make the product a perfect cube. Further find the cube root of the product.

50.The three numbers are in the ration 2:3:4. The sum of their cubes is 33957. Find the numbers.

51.The volume of a cube in 9261000 m³. Find the side of the cube.

52.(i) Three numbers are in the ratio 1:2:3. The sum of their cubes is 98784. Find the numbers.

53.(i) Divide the number 26244 by the smallest so that the quotient is a perfect cube. Also find the cube root of number obtained.

CUBES AND CUBE ROOTS

CONTENTS

• Cube
• Some Interesting Patterns
• Cube Root
• Digits in cube root of a Number
• Sum of Numbers

CUBES

A cube is a solid figure which has all its sides equal. If side of a cube is 1 cm then 27 such cubes can make a big cube of side 3 cm. So, no. 1, 8, 27, 64, …. are called perfect cube numbers.

Table–1
Number | Cube
1 | \( 1^3 = 1 \)
2 | \( 2^3 = 8 \)
3 | \( 3^3 = 27 \)
4 | \( 4^3 = 64 \)
5 | \( 5^3 = 125 \)
6 | \( 6^3 = 216 \)
7 | \( 7^3 = 343 \)
8 | \( 8^3 = 512 \)
9 | \( 9^3 = 729 \)
10 | \( 10^3 = 1000 \)

There are only ten perfect cubes from 1 to 1000 and four from 1 to 100 .

Table–2
Number | Cube
11 | 1331
12 | 1728
13 | 2197
14 | 2744
15 | 3375
16 | 4096
17 | 4913
18 | 5832
19 | 6859
20 | 8000

Results :

• 1. Cube of even number is also an even number.
• 2. Cube of an odd number is also an odd number.
• 3. Unit place of cube of a number whose unit digit is 2, 3, 7, 8 is 8, 7, 3, 2 respectively

SOME INTERESTING PATTERNS

1. Adding consecutive odd numbers :
Observe the following pattern of sums of odd numbers.
\( 1 = 1 = 1^3 \)
\( 3 + 5 = 8 = 2^3 \)
\( 7 + 9 + 11 = 27 = 3^3 \)
\( 13 + 15 + 17 + 19 = 64 = 4^3 \)
\( 21 + 23 + 25 + 27 + 29 = 125 = 5^3 \)

Question. How many consecutive odd numbers will be needed to obtain the sum as \( 10^3 \) ?
Answer: 10 (91, 93, 95, 97, 99, 101, 103, 105, 107, 109)

2. Prime factors of perfect cube :

Each prime number appears three or multiple of 3 times in its cube.
Eg. 8 = 2 × 2 × 2
Eg. 64 = (2 × 2) × (2 × 2) × (2 × 2)
Eg 125 = (5 × 5 ×5) = \( 5^3 \) = perfect cube number
\(\therefore\) \( a^3 \) is a perfect cube number.

Question. Is 128 a perfect cube ?
Answer: 128 = (2 × 2) × (2 × 2) × (2 × 2) × 2 = \( 2^7 \). \(\because\) power of 2 is not a multiple of 3. \(\therefore\) it is not a perfect cube.

Question. Find the cubes of the following numbers:
(a) 2 (b) 3 (c) 7 (d) 0.9 (e) (–5) (f) – 0.1
Answer: (a) 2 × 2 × 2 = 8 \(\Rightarrow\) \( 2^3 = 8 \)
(b) 3 × 3 × 3 = 27 \(\Rightarrow\) \( 3^3 = 27 \)
(c) 7 × 7 × 7 = 343 \(\Rightarrow\) \( 7^3 = 343 \)
(d) 0.9 × 0.9 × 0.9 = 0.729 \(\Rightarrow\) \( (0.9)^3 = 0.729 \)
(e) (–5) × (–5) × (–5) = –125 \(\Rightarrow\) \( (–5)^3 = –125 \)
(f) (–0.1) × (–0.1) × (–0.1) = –0.001 \(\Rightarrow\) \( (–0.1)^3 = –0.001 \)
A natural number is said to be a perfect cube if it is the cube of another natural number. We know that when odd number of negative factors are multiplied, the product is always negative, so cube can be negative also.

CUBE ROOT

If \( 2^2 = 4 \), then the square root of 4, i.e., \( \sqrt{4} = 2 \). Similarly, if \( 2^3 = 8 \), then the cube root of 8 is 2. It is written as \( \sqrt[3]{8} = 2 \). If \( 3^3 = 27 \), then the cube root of 27 is 3. Thus, \( \sqrt[3]{27} = 3 \). Note that the symbol \( \sqrt{} \) implied square root. For our convenience, we omit 2 from \( \sqrt[2]{} \). But for a cube root, we should use the symbol \( \sqrt[3]{} \), and it cannot be omitted also we can use \( ( )^{1/3} \) for cube root.

Prime Factorisation Method for Finding the Cube Root

Let us take some examples here

Question. Find the cube root of 1728.
Answer: \( \sqrt[3]{1728} = (1728)^{1/3} \).
Step : 1 First factorise the given number into its prime factors. \( \sqrt[3]{1728} = \sqrt[3]{2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3} \)
Step : 2 Then group the factors in 3s. \( \sqrt[3]{1728} = \sqrt[3]{2^3 \times 2^3 \times 3^3} \)
Step : 3 Take one prime factor from each group. \( \sqrt[3]{1728} = 2 \times 2 \times 3 = 12 \). \(\therefore\) \( \sqrt[3]{1728} = 12 \)

Question. Find the value of \( \sqrt[3]{216} \)
Answer: \( \sqrt[3]{216} = (216)^{1/3} \).
Step-1: Factorise the given number into its prime factors. \( \sqrt[3]{216} = \sqrt[3]{2\times 2\times 2\times 3\times 3\times 3} \)
Step-2: Group the factors in 3s. \( \sqrt[3]{216} = \sqrt[3]{2^3 \times 3^3} \)
Step-3 : Take one prime factor from each group. \( \sqrt[3]{216} = 2 \times 3 = 6 \). \(\therefore\) \( \sqrt[3]{216} = 2 \times 3 = 6 \)

Observe \( 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125, \dots \)
All cubes of even numbers are even and cubes of odd numbers are odd. Cubes of negative numbers are negative.

Question. Find the cube root of 46656.
Answer: (i) The unit digit of the number is 6, so the cube root will also have 6 in the unit digit. (ii) Separate the number as 46 656. 46 is greater than \( 3^3 \) but less than \( 4^3 \), so the tens digit is 3. (iii) The required number is 36.

Question. Find the cube root of 195112.
Answer: (i) Unit digit of the given number is 2, so the required number has unit digit 8. (ii) 195 112, so 195 > \( 5^3 \) but < \( 6^3 \). So, required number is 58.
Note : Above method works for perfect cube numbers called cube root by approximation.

DIGITS IN CUBE ROOT OF A NUMBER

Use dots on digit of given number starting from unit digit & leaving 2 next digits, now digits in cube root is same as the sum of dots.

Question. Find the digits in cube root of the following numbers.
(i) 1728 (ii) 175616 (iii) 8 (iv) 97336 (v) 9261 (vi) 68921000
Answer: (i) 1728: two dots \(\therefore\) 2 digits in cube root
(ii) 175616: two dots \(\therefore\) 2 digits in cube root
(iii) 8: Only one dot \(\therefore\) 1 digit in cube root
(v) 9261: Two dots \(\therefore\) 2 digit in cube root
(vi) 68921000: Three dots \(\therefore\) 3 digit in cube root

SUM OF NUMBERS

The sum of first ‘n’ natural numbers.
\( 1 + 2 + 3 + ....... + n = \frac{n(n + 1)}{2} \)

Question. Find sum of first 6 natural numbers.
Answer: n = 6 \(\therefore\) Sum = \( \frac{6(6 + 1)}{2} = 3 \times 7 = 21 \)

Question. Find sum of 10 + 11 + ....... + 20.
Answer: \(\because\) Sum of 1 to 20 is \( \frac{20(20 + 1)}{2} = 10 \times 21 = 210 \) and sum of 1 to 9 is \( \frac{9(9 + 1)}{2} = \frac{9 \times 10}{2} = 45 \). \(\therefore\) 10 + 11 + ....... + 20 = 210 – 45 = 165

The sum of Square of first ‘n’ natural numbers.
\( 1^2 + 2^2 + 3^2 + .......... + n^2 = \frac{n(n + 1)(2n + 1)}{6} \)

Question. Find sum of squares of first five natural numbers.
Answer: \( 1^2 + 2^2 + 3^2 + 4^2 + 5^2 \). \(\therefore\) n = 5 \(\therefore\) sum = \( \frac{5(5 + 1)(10 + 1)}{6} = 55 \)

The sum of cube of first ‘n’ natural numbers.
\( 1^3 + 2^3 + 3^3 + 4^3 + ............ + n^3 = \left[ \frac{n(n + 1)}{2} \right]^2 \)

Question. Find sum of cube of first five natural numbers.
Answer: \( 1^3 + 2^3 + .......... + 5^3 \). \(\therefore\) n = 5 \(\therefore\) sum = \( \left[ \frac{5(5 + 1)}{2} \right]^2 = (5 \times 3)^2 = 225 \)

EXERCISE

Question. Find the number of digits in the cube root if number of digits in perfect cube numbers as follows.
(i) 6 (ii) 5 (iii) 4 (iv) 3 (v) 2 (vi) 1 (vii) 7
Answer: (i) 2 ; (ii) 2 ; (iii) 2 ; (iv) 1 ; (v) 1 ; (vi) 1; (vii) 3

Question. Find the value of \( \sqrt{117 + \sqrt[3]{19683}} \).
Answer: \( \sqrt{117 + 27} = 12 \)

Question. Which of the following are perfect cube ?
(i) 10 (ii) 100 (iii) 1000 (iv) \( 10^4 \) (v) \( 10^5 \) (vi) \( 10^6 \)
Answer: (iii), (vi)

Question. Find the value of \( \frac{(2)^3 + (10)^3}{\sqrt{1016064}} \)
Answer: 1

Question. Find the sum of cubes of first 10 natural numbers.
Answer: 3025

Question. Find the value of \( (1^3 + 2^3 + 3^3 + 4^4 + ....+ 15^3) – (1^2 + 2^2 + 3^2 + .....+ 10^2) \)
Answer: 14015

Question. Find the cube root of the following numbers by inspection.
(i) 12167 (ii) 46.656 (iii) 6859 (iv) 912673 (v) 29791
Answer: (i) 23 ; (ii) 3.6; (iii) 19; (iv) 97; (v) 31

Question. Find cube root of \( [ 5 \sqrt{100} + \sqrt{49} + (79507)^{1/3} ] \).
Answer: 10

Question. Find cube root by prime factorisation
(i) 4913 (ii) 13824 (iii) 175616 (iv) 456533
Answer: (i) 17; (ii) 24; (iii) 56; (iv) 77

Question. Find the least number by which when multiply the following numbers, such that the number become perfect cube.
(i) 2048 (ii) 1029 (iii) 45 (iv) 5832
Answer: (i) 2; (ii) 9; (iii) 75; (iv) 1

Question. Find the least number by which when divide the following numbers. The number become perfect cube also find cube root of new number
(i) 4394 (ii) 8575 (iii) 7986 (iv) 28672
Answer: (i) 2, 13; (ii) 25, 7; (iii) 6, 11; (iv) 7, 16