NCERT Solutions for Class 8 Maths Chapter 7 - Embibe
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# NCERT Solutions for Class 8 Maths Chapter 7: Cubes & Cube Roots Embibe’s solutions for Cubes and Cube Roots will be very helpful for students to score good marks in the final exam. As it is one of the most important chapters for Class 8 Maths, we have provided detailed NCERT Solutions for Class 8 Maths Chapter 7 for all the intext and exercise questions. The answers to all problems within this chapter are presented in an easy-to-understand and detailed manner to help the students understand more effectively.

With the help of these Class 8 Maths solutions for Chapter 7, students will be able to clear all their doubts related to Cubes and Cube Roots. A team of academic experts have prepared these solutions and presented them in a step-by-step manner. The solutions will definitely help students strengthen their concepts for this chapter. It will also help them to complete their assignments and homework on time. Continue reading to know more.

## NCERT Solutions for Class 8 Maths Chapter 7: Cubes and Cube Roots

NCERT Solutions for Class 8 Maths Chapter 7 is must for all students to obtain an excellent score in the final exam. Before we present you the solutions, let’s have an overview of the topics and sub-topics under Class 8 Maths – Cube & Cube Roots chapter. Click on any topic to download its solutions as a PDF.

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### NCERT Solutions for Class 8 Maths Chapter 7: Solved Exercises and In-Text Questions

In this section, we have provided solutions for 8th Class Maths Chapter 7 for all exercises and intext questions. You can view the solutions here on this page or download the complete solutions as PDF. To download the PDF, just click on the link provided at the end of these solutions. With the help of solutions provided for Class 8 Maths book exercises, students can also complete their class and home assessment on time without any difficulty.

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### NCERT Solutions for Class 8 Maths Chapter 7: Chapter Summary

What is Hardy – Ramanujan Number?

The smallest number that can be expressed as a sum of two cubes in two different ways is known as the Hardy – Ramanujan Number.

1729 is the smallest such type of number and hence is called the Hardy – Ramanujan Number.

1729 = 1728 + 1 = 123 + 13

1729 = 1000 + 729 = 103 + 93

There are many such numbers such as 4104 (2, 16; 9, 15), 13832 (18, 20; 2, 24), etc.

What is Cube of a number?

Cube is a number which is obtained when a number is multiplied by taking it three times. We can also say that the cube is that number raised to a power of 3. Numbers like 1, 8, 27 are known as perfect cubes as they can be expressed as a=b3. Here, a= perfect cube and b= number

 Number Cube Number 1 13= 1 2 23= 8 3 33= 27 4 43= 64

Properties of Cube Numbers:

1. The cube of an even natural number is even.
2. The cube of an odd natural number is odd.
3. The cube of a negative number is always negative.

Note:

→There are only 10 perfect cubes between 1 and 1000.

 Number Cubes 1 13= 1 2 23= 8 3 33= 27 4 43= 64 5 53= 125 6 63= 216 7 73= 343 8 83= 512 9 93= 729 10 103= 1000

What is Cube Root?

Cube root is the inverse operation of finding cube. It is denoted by the symbol : ∛

Read the following statements to further understand the meaning of cube root.

We know that, 23=8. Here we can say that cube root of 8 is 2.

It can also be written as ∛ 8 = 2

We can find cube root of a given number by Prime Factorisation Method. For example: consider a number 3375. It’s prime factors are 3375= 3 x 3 x 3 x 5 x 5 x 5 = 33 x 53 = (3 x 5)3.

### FAQs

Here we have provided some of the frequently asked questions related to this chapter:

Q1: State true or false.
(i) Cube of any odd number is even.
(ii) A perfect cube does not end with two zeros.
(iii) If the cube of a number ends with 5, then its cube ends with 25.
(iv) There is no perfect cube which ends with 8.
(v) The cube of a two-digit number may be a three-digit number.
(vi) The cube of a two-digit number may have seven or more digits.
(vii) The cube of a single-digit number may be a single-digit number.

A: i) False
ii) True
iii) False
iv) False
v) False
vi) False
vii) True

Q2: Find the cube root of each of the following numbers by prime factorisation method.
(i) 64
(ii) 512
(iii) 10648
(iv) 27000

A: i) 64 = 2×2×2×2×2×2
By grouping the factors in triplets of equal factors, 64 = (2×2×2)×(2×2×2)
Here, 64 can be grouped into triplets of equal factors,
∴ 64 = 2×2 = 4
Hence, 4 is the cube root of 64.
ii) 512 = 2×2×2×2×2×2×2×2×2
By grouping the factors in triplets of equal factors, 512 = (2×2×2)×(2×2×2)×(2×2×2)
Here, 512 can be grouped into triplets of equal factors,
∴ 512 = 2×2×2 = 8
Hence, 8 is the cube root of 512.
iii) 10648 = 2×2×2×11×11×11
By grouping the factors in triplets of equal factors, 10648 = (2×2×2)×(11×11×11)
Here, 10648 can be grouped into triplets of equal factors,
∴ 10648 = 2 ×11 = 22
Hence, 22 is the cube root of 10648.
iv) 27000 = 2×2×2×3×3×3×3×5×5×5
By grouping the factors in triplets of equal factors, 27000 = (2×2×2)×(3×3×3)×(5×5×5)
Here, 27000 can be grouped into triplets of equal factors,
∴ 27000 = (2×3×5) = 30
Hence, 30 is the cube root of 27000.

Q3: You are told that 1,331 is a perfect cube. Can you guess without factorisation what is its cube root? Similarly, guess the cube roots of 4913, 12167, 32768.

A: (i) By grouping the digits, we get 1 and 331
We know that, since the unit digit of a cube is 1, the unit digit of cube root is 1.
∴ We get 1 as unit digit of the cube root of 1331.
The cube of 1 matches with the number of the second group.
∴ The ten’s digit of our cube root is taken as the unit place of the smallest number.
We know that, the unit’s digit of the cube of a number having digit as unit’s place 1 is 1.
∴ ∛1331 = 11
(ii) By grouping the digits, we get 4 and 913
We know that, since, the unit digit of cube is 3, the unit digit of cube root is 7.
∴ we get 7 as unit digit of the cube root of 4913. We know 13 = 1 and 23 = 8 , 1 > 4 > 8.
Thus, 1 is taken as ten digit of cube root.
∴ ∛4913 = 17
(iii) By grouping the digits, we get 12 and 167.
We know that, since, the unit digit of cube is 7, the unit digit of cube root is 3.
∴ 3 is the unit digit of the cube root of 12167 We know 23 = 8 and 33 = 27 , 8 > 12 > 27.
Thus, 2 is taken as ten digit of cube root.
∴ ∛12167= 23
(iv) By grouping the digits, we get 32 and 768.
We know that, since, the unit digit of cube is 8, the unit digit of cube root is 2.
∴ 2 is the unit digit of the cube root of 32768. We know 33 = 27 and 43 = 64 , 27 > 32 > 64.
Thus, 3 is taken as ten digit of cube root.
∴ ∛32768= 32

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