NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

Mathematics (Maths) Class 8

Created by: Full Circle

Class 8 : NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

The document NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev is a part of the Class 8 Course Mathematics (Maths) Class 8.
All you need of Class 8 at this link: Class 8

Exercise 7.1

Question 1. Which of the following numbers are not perfect cubes?

(i) 216 (ii) 128 (iii) 1000
(iv) 100 (v) 46656

Solution: 

(i) We have 216 = 2 * 2 * 2 * 3 * 3 * 3

Grouping the prime factors of 216 into triples, no factor is left over.

∴ 216 is a perfect cube.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(ii) We have 128 = 2 * 2 * 2 * 2 * 2 * 2 * 2

Grouping the prime factors of 128 into triples, we are left over with 2 as ungrouped factor.

∴ 128 is not a perfect cube.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(iii) We have 1000 = 2 * 2 * 2 * 5 * 5 * 5

Grouping the prime factors of 1000 into triples, we are not left over with any factor.

∴ 1000 is a perfect cube.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(iv) We have 100 = 2 * 2 * 5 * 5

Grouping the prime factors into triples, we do not get any triples. Factors 2 * 2 and 5 * 5 are not in triples.

∴ 100 is not a perfect cube.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(v) We have 46656 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 3 * 3 * 3

Grouping the prime factors of 46656 in triples we are not left over with any prime factor.

∴ 46656 is a perfect cube.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

Question 2. Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.

(i) 243 (ii) 256 (iii) 72
(iv) 675 (v) 100

Solution: 

(i) We have 243 = 3 * 3 * 3 * 3 * 3

The prime factor 3 is not a group of three.

∴ 243 is not a perfect cube.

Now, [243] * 3 = [3 * 3 * 3 * 3 * 3] * 3

or 729 = 3 * 3 * 3 * 3 * 3 * 3

Now, 729 becomes a perfect cube.

Thus, the smallest required number to multiply 243 to make it a perfect cube is 3.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(ii) We have 256 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2

Grouping the prime factors of 256 in triples, we are left over with 2 * 2.

∴ 256 is not a perfect cube.

Now, [256] * 2 = [2 * 2 * 2 * 2 * 2 * 2 * 2 * 2] * 2

or 512 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 i.e. 512 is a perfect cube.

Thus, the required smallest number is 2.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(iii) We have 72 = 2 * 2 * 2 * 3 * 3

Grouping the prime factors of 72 in triples, we are left over with 3 * 3.

∴ 72 is not a perfect cube.

Now, [72] * 3 = [2 * 2 * 2 * 3 * 3] * 3

or 216 = 2 * 2 * 2 * 3 * 3 * 3

i.e. 216 is a perfect cube.

∴ The smallest number required to multiply 72 to make it a perfect cube is 3.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(iv) We have 675 = 3 * 3 * 3 * 5 * 5

Grouping the prime factors of 675 to triples, we are left over with 5 * 5.

∴ 675 is not a perfect cube.

Now, [675] * 5 = [3 * 3 * 3 * 5 * 5] * 5

or 3375 = 3 * 3 * 3 * 5 * 5 * 5

Now, 3375 is a perfect cube.

Thus, the smallest required number to multiply 675 such that the new number is a perfect cube is 5.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(v) We have 100 = 2 * 2 * 5 * 5

The prime factor are not in the groups of triples.

∴ 100 is not a perfect cube.

Now [100] * 2 * 5 = [2 * 2 * 5 * 5] * 2 * 5

or [100] * 10 = 2 * 2 * 2 * 5 * 5 * 5

1000 = 2 * 2 * 2 * 5 * 5 * 5

Now, 1000 is a perfect cube.

Thus, the required smallest number is 10.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

Question 3. Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.

(i) 81 (ii) 128 (iii) 135
(iv) 192 (v) 704

Solution:

(i) We have 81 = 3 * 3 * 3 * 3

Grouping the prime factors of 81 into triples, we are left with 3.

∴ 81 is not a perfect cube.

Now, [81] /3 = [3 * 3 * 3 * 3] ÷ 3

or  27 = 3 * 3 * 3

i.e. 27 is a prefect cube

Thus, the required smallest number is 3.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(ii) We have 128 = 2 * 2 * 2 * 2 * 2 * 2 * 2

Grouping the prime factors of 128 into triples, we are left with 2.

∴ 128 is not a perfect cube

Now, [128] /2 = [2 * 2 * 2 * 2 * 2 * 2 * 2] ÷2

or  64 = 2 * 2 * 2 * 2 * 2 * 2

i.e. 64 is a perfect cube.

∴ The smallest required number is 2.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(iii) We have 135 = 3 * 3 * 3 * 5

Grouping the prime factors of 135 into triples, we are left over with 5.

∴ 135 is not a perfect cube

Now, [135] /5 = [3 * 3 * 3 * 5] ÷5

or 27 = 3 * 3 * 3

i.e. 27 is a perfect cube.

Thus, the required smallest number is 5.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(iv) We have 192 = 2 * 2 * 2 * 2 * 2 * 2 * 3

Grouping the prime factors of 192 into triples, 3 is left over.

∴ 192 is not a perfect cube.

Now, [192] ∏ 3 = [2 * 2 * 2 * 2 * 2 * 2 * 3] ÷3

or  64 = 2 * 2 * 2 * 2 * 2 * 2

i.e. 64 is a perfect cube.

Thus, the required smallest number is 3.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

(v) We have 704 = 2 * 2 * 2 * 2 * 2 * 2 * 11

Grouping the prime factors of 704 into triples, 11 is left over.

∴ [704] /11 = [2 * 2 * 2 * 2 * 2 * 2 * 11] ÷11

or 64 = 2 * 2 * 2 * 2 * 2 * 2

i.e. 64 is a perfect cube.

Thus, the required smallest number is 11.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

Question 4. Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will he need to form a cube?

Solution: Sides of the cuboid are: 5 cm, 2 cm, 5 cm

∴ Volume of the cuboid = 5 cm * 2 cm * 5 cm

To form it as a cube its dimension should be in the group of triples.

∴ Volume of the required cube = [5 cm * 5 cm * 2 cm] * 5 cm * 2 cm * 2cm = [5 * 5 * 2 cm3] = 20 cm3

Thus, the required number of cuboids = 20

Cube Roots

Finding cube root is the inverse operation of finding cube.

Since, 4= 64, so the cube root of 64 is 4.

The symbol for cube root is ∛

E*amples: 

23 = 8 → ∛8 = 2

33 = 27 → ∛27 = 3

4= 64 →∛64 = 4

53 = 125 → ∛125 = 5

63 = 216 → ∛ 216 = 6   

73 = 343 → ∛343 = 7

Cube Root of a Cube Number through Estimation

Example 1. Find the cube root of 614125 through estimation. 

Solution: We use the following steps to find the cube root through estimation. 

I. Form two groups three digits each starting from the right most”

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

II. 1st group (125) gives us the unit’s digit of cube root.

∵ Cube of a number ending in 5, also ends in 5.

∴ Unit’s digit of the cube root is 5.

III. 2nd group (here 614) gives us the ten’s digit of the cube root.

Since, 83 = 512 and 93 = 729

Also 512 < 614 < 729

∴ We guess the ten’s digit of the cube root with the help of unit’s digit of 512.

We know that, if a number ends in 8, its cube will end in 2.

∴ Ten’s digit of the required cube root must be 8.

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

Cube Root through Prime Factorisation Method

Example 1. Find the cube root of 1728.

Solution: By prime factorisation, we have

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

Example 2. Find the cube root of 27000 by prime factorisation.

Solution:

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

Question: State true or false: for any integer m, m2 < m3. Why?

Solution: It is not always true.

For example, let m = –1

We have m2 = (–1)2 = 1

and m3 = (–1)3 = –1

∴ The above statement, i.e. m2 < m3 is false.

Share with a friend

Complete Syllabus of Class 8

Dynamic Test

Content Category

Related Searches

Sample Paper

,

Semester Notes

,

Viva Questions

,

MCQs

,

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

,

Exam

,

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

,

Extra Questions

,

shortcuts and tricks

,

mock tests for examination

,

video lectures

,

NCERT Solutions(Part- 2)- Cubes and Cube Roots Class 8 Notes | EduRev

,

Objective type Questions

,

Previous Year Questions with Solutions

,

pdf

,

Important questions

,

Summary

,

practice quizzes

,

Free

,

past year papers

,

ppt

,

study material

;