NCERT Solutions for Class 8 Maths Chapter 7 Cubes and Cube Roots: The calculation is the backbone of mathematics and if you want to have a good command over mathematics then you can’t neglect the calculation part. It's not just about calculating right but right calculation in minimal time using calculations technique. In this article, you will get CBSE NCERT solutions for class 8 maths chapter 7 cubes and cube roots. Cube is a well-known shape in geometry that comes with 3- dimensions that have all sides equal but here we will not discuss that cube. In this chapter, you are going to learn how cubes and cube roots are calculated for a given number. In solutions of NCERT for class 8 maths chapter 7 cubes and cube roots, you will get short tricks and techniques to crack the mathematical problems in minimal time. It will enhance your calculation skill which is very important for many competitive exams like SSC.
Cube-
Cube means a number will be multiplied 3-times by itself. For example:- If we want to calculate the cube of 3 then the cube will be 3×3×3 = 27, similarly, a cube of 5 is 5×5×5=125.
Cube Root-
Cube root then cube root is just a reverse application of cube means when a number multiplied 3-times by itself then it will result in the cube and the root number of this result is called cube root. For example:- Cube of 3 = 3×3×3 which is equal to 27 and similarly in the reverse manner cube root of 27 is equal to 3. There is another way to understand this, Which number we should multiply 3 times by itself to gain 27, then the number which will strike in your mind will be 3.
There are 2 exercises with 7 questions in this chapter. You will get detailed explanations of all these questions in the CBSE NCERT solutions for class 8 maths chapter 7 cubes and cube roots. Check NCERT solutions for class 6 to 12 by clicking on the above link.
7.1 Introduction
7.2 Cubes
7.2.1 Some interesting patterns
7.2.2 Smallest multiple that is a perfect cube
7.3 Cube Roots
7.3.1 Cube root through prime factorization method
7.3.2 Cube root of a cube number
Q(i) Find the one’s digit of the cube of each of the following numbers.
The detailed solution for the above-mentioned question is as follows,
Since the given number ends with 1, so the one’s digit of the cube of 3331 will be 1.
Q(ii) Find the one’s digit of the cube of each of the following numbers.
The detailed solution for the above-mentioned question is as follows
Since the given number ends with 8, so the one’s digit of the cube of 8888 will be 2.
The detailed solution for the above-mentioned question is as follows,
Since the given number has 9 at units place, so the one’s digit of the cube of 149 will be 9.
Q(iv) Find the one’s digit of the cube of each of the following numbers.
The detailed solution for the above-mentioned questions is as follows
Since the given number ends with 5, so one's digit of its cube will also end with 5.
Q(v) Find the one’s digit of the cube of each of the following numbers.
The solution to the above-mentioned question is as follows,
The given digit is ending with 4. So the one’s digit of the cube of 1024 will be 4.
Q(vi) Find the one’s digit of the cube of each of the following numbers.
The detailed solution for above-mentioned question is as follows,
The given number is ending with 7, so its cube will end with 3.
Q(vii) Find the one’s digit of the cube of each of the following numbers.
The detailed solution for the above-mentioned question is as follows,
Since the given number ends with 2, so its cube will end with 8.
Q(viii) Find the one’s digit of the cube of each of the following numbers.
The detailed solution for the above-mentioned question is as follows,
Since the given number has 3 at units place, so, its cube will end with 7.
Q(a) Express the following numbers as the sum of odd numbers using the above pattern?
The detailed solution for the above mentioned question is as follows,
216 => 31 + 33 + 35 + 37 + 39 + 41
Q(b) Express the following numbers as the sum of odd numbers using the above pattern?
The detailed solution for the above-mentioned question is as follows
512 => 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71
Q(c) Express the following numbers as the sum of odd numbers using the above pattern?
The detailed solution for the above-mentioned question is as follows
7^{3 }= 43 + 45 + 47 + 49 + 51 + 53 + 55
find the value of the following.
The value of the following question is:
Q(iii). Consider the following pattern.
Using the above pattern,find the value of the following.
The detailed solution for the above-written question is mentioned below,
Q(iv) Consider the following pattern.
Using the above pattern, find the value of the following.
The detailed solution for the above written question is mentioned below
Q. Which of the following are perfect cubes?
We will find it by prime factorization whether they make a pair of three prime numbers or not.
(1) . So not a perfect cube.
(2) . So it is a perfect cube.
(3) . So it is a perfect cube.
(4) . So it is a perfect cube.
(5) . So it is not a perfect cube.
(6) . So it is a perfect cube.
(7) . So it is not a perfect cube.
(8) . So it is a perfect cube.
Q. Check which of the following are perfect cubes.
What pattern do you observe in these perfect cubes?
The detailed solution for the above-written question is as follows
By prime factorization:
(i) . So it is not a perfect cube.
(ii) . So it is not a perfect cube.
(iii) . So it is a perfect cube.
(iv) . So it is not a perfect cube.
(v) . So it is a perfect cube.
(vi) . So it is not a perfect cube.
(vii) . So it is not a perfect cube.
(viii) . So it is not a perfect cube.
(ix) . So it is a perfect cube.
(x) . So it is a perfect cube.
We observe that the numbers above which are perfect cube have the number of zeros in multiple of 3.
NCERT solutions for class 8 maths chapter 7 cubes and cube roots-Exercise: 7.1
Q.1(i) Which of the following numbers are not perfect cubes?
The detailed solution for the above-written question is as follows
By prime factorization of 216 gives:
Since prime numbers are present in pairs of three, so the given number is a perfect cube.
Q.1(ii) Which of the following numbers are not perfect cubes?
We have 128. By prime factorization we get,
Since the prime numbers are not in pairs of three, so the given number is not a perfect cube.
Q.1(iii) Which of the following numbers are not perfect cubes?
The detailed solution for the above written is as follows
By prime factorization of 1000 we get :
.
So the given number is a perfect cube.
Q.1(iv) Which of the following numbers are not perfect cubes?
The detailed solution for the above-written question is as follows
By prime factorization of 100 :
.
Since prime numbers are not in pair of three so given number is not a perfect cube.
Q.1(v) Which of the following numbers are not perfect cubes?
We have 46656, by prime factorisation:
.
Since prime numbers are in group of three. So the given number is a perfect cube.
This can be found by knowing about the prime factors of the number.
(i) 243 : .
So it must be multiplied by 3.
(ii) 256 :
So the given number must be multiplied by 2 to make it a perfect cube.
(iii) 72 :
So 72 must be multiplied by 3 to make it a perfect cube.
(iv) 675 :
So it should be multiplied by 5.
(v) 100 :
So it should be multiplied by 10.
By prime factorization of given numbers :
(i) 81 :
So given number needs to be divided by 3 to get a perfect cube.
(ii) 128 : .
So the given number needs to be divided by 2 to get a perfect cube.
(iii) 135 :
So the given number needs to be divided by 5 to get a perfect cube.
(iv) 192 :
So the given number needs to be divided by 3 to get a perfect cube.
(v) 704 :
So the given number needs to be divided by 11 to get a perfect cube.
Q.4 Parikshit makes a cuboid of plasticine of sides . How many such cuboids will he need to form a cube?
Answer:
Volume of cuboid is
To make it a cube need to make this a pefect cube number.
So we need cuboids
or 20 cuboids.
The detailed solution for the above-written question is as follows.
False.
or
or
or
Now put any number less than 1, we see that this relation doesn't hold.
So for m<1 this condition is not true.
NCERT solutions for class 8 maths chapter 7 cubes and cube roots-Exercise: 7.2
Q.1(i) Find the cube root of each of the following numbers by prime factorisation method.
The detailed solution for the above-written question is as follows
Prime factorization of 64 gives :
So its cube root is = 4
Q.1(ii) Find the cube root of each of the following numbers by prime factorisation method.
By prime factorisation of 512 :
So its cube root is
Q.1(iii) Find the cube root of each of the following numbers by prime factorisation method.
The detailed solution for the above-written question is as follows
Prime factorization of 10648 gives :
So its cube root is 22.
Q.1(iv) Find the cube root of each of the following numbers by prime factorisation method.
The detailed solution for the above-written question is as follows
By prime factorization method, we get :
So its cube root is 30.
The detailed solution for the above-written question is as follows
By prime factorization:
So its cube root is 25.
Q.1(vi) Find the cube root of each of the following numbers by prime factorisation method.
The detailed solution for the above-written question is as follows
By prime factorization:
So its cube root is 24.
Q.1(vii) Find the cube root of each of the following numbers by prime factorisation method.
The detailed solution for the above-written question is as follows
By prime factorization:
So its cube root is = 48.
Q.1(viii) Find the cube root of each of the following numbers by prime factorisation method.
The detailed solution for the above-written question is as follows
By prime factorization, we get :
So its cube root is = 36.
Q.1(ix) Find the cube root of each of the following numbers by prime factorisation method.
The detailed solution for the above-written question is as follows
By prime factorization we get :
So its cube root is = 56.
Q.1(x) Find the cube root of each of the following numbers by prime factorisation method.
The detailed solution for the above-written question is as follows
By prime factorization, we get :
So its cube root is = 45.
(i) Cube of any odd number is even.
(ii) A perfect cube does not end with two zeros.
(iii) If square 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.
(i) False. Cube of an odd number can never be even.
(ii) True. Perfect cube number ends with zeros multiple of three.
(iii) False. We can say only about units place.
(iv) False. Cube of numbers which ends with 2 end with 8.
(v) False. Can never be.
(vi) False. Can never be. It can be proved by taking examples.
(vii) True. e.g. 1,2
We have 1331.
Divide number in two parts: The first part is 1 and second is 331.
Since the given number is ending with 1 so the last digit of cube root will be 1.
In the first part, we have 1.
So
By estimation, the cube root of 1331 is 11.
Similarly for all other parts.
4913:- First part is 4 and the second part is 913.
The number is ending with 3 so its cube root will have 7 at units place.
In the first part, the nearest cube root is 1.
So the cube root of 4913 is 17.
12167:- First part is 12 and the second part is 167.
The number is ending with 7 so its cube root will have 3 at units place.
In the first part, the nearest cube root is 2.
So the cube root of 12167 is 23.
32768:- First part is 32 and the second part is 768.
The number is ending with 8, so its cube root will have 2 at units place.
In the first part, the nearest cube root is 3.
So the cube root of 32768 is 32.
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