RBSE Solutions Class 8 Maths Chapter 2 Cube and Cube Roots Important Questions

I. Objective Type Questions

Question 1. How many consecutive odd numbers will be required to get \( 10^3 \)?
(a) 5
(b) 8
(c) 10
(d) 100
Answer: (c) 10
In simple words: To find the cube of a number, we can add a certain number of consecutive odd numbers. For \( 10^3 \), we need 10 consecutive odd numbers to sum up to 1000. For example, \( 1^3=1 \) (1 odd number), \( 2^3=8 \) (\( 3+5 \), 2 odd numbers), \( 3^3=27 \) (\( 7+9+11 \), 3 odd numbers). This pattern helps to understand the relationship between cubes and sums of odd numbers.

🎯 Exam Tip: Remember the pattern: \( n^3 \) is the sum of 'n' consecutive odd numbers. The first odd number in the sequence for \( n^3 \) is \( n^2 - n + 1 \).

Question 2. How many perfect cube numbers are there in between 1 to 1000?
(a) 10
(b) 18
(c) 25
(d) 52
Answer: (a) 10
In simple words: A perfect cube number is a number you get by multiplying an integer by itself three times. We need to count all the whole numbers whose cubes are between 1 and 1000 (including 1000, but usually not 1). These are \( 1^3=1 \), \( 2^3=8 \), \( 3^3=27 \), \( 4^3=64 \), \( 5^3=125 \), \( 6^3=216 \), \( 7^3=343 \), \( 8^3=512 \), \( 9^3=729 \), and \( 10^3=1000 \). Counting these gives us 10 numbers.

🎯 Exam Tip: When a question asks "between X to Y", it usually implies including X and Y if they are perfect cubes themselves. Always list the cubes to be sure, starting from \( 1^3 \) upwards.

Question 3. The cube of 7 is
(a) 49
(b) 243
(c) 343
(d) 21
Answer: (c) 343
In simple words: To find the cube of 7, you multiply 7 by itself three times: \( 7 \times 7 \times 7 \). First, \( 7 \times 7 = 49 \). Then, \( 49 \times 7 = 343 \). The result is 343.

🎯 Exam Tip: It is helpful to memorize the cubes of numbers from 1 to at least 10 for quick calculations in exams.

Question 4. What is the cube root of 13824?
(a) 24
(b) 56
Answer: (a) 24
In simple words: To find the cube root, we look for a number that, when multiplied by itself three times, gives 13824. Since 13824 ends in 4, its cube root must also end in 4. The number 24 fits this, and \( 24 \times 24 \times 24 = 13824 \). Knowing the unit digit helps narrow down the choices quickly.

🎯 Exam Tip: For cube roots, examine the unit digit of the number: if it ends in 4, its cube root ends in 4. Also, estimate the range: \( 20^3=8000 \) and \( 30^3=27000 \), so the answer must be between 20 and 30.

Question 6. Cube root - 216 is
(a) 6
(b) 16
(c) - 6
(d) - 16
Answer: (c) - 6
In simple words: The cube root of a negative number is always negative. We know that \( 6 \times 6 \times 6 = 216 \). Therefore, \( (-6) \times (-6) \times (-6) = -216 \). So, the cube root of -216 is -6.

🎯 Exam Tip: Remember that an odd power (like cube) of a negative number remains negative, while an even power (like square) of a negative number becomes positive.

Question 7. Cube of 80 is
(a) 51200
(b) 512000
(c) 512
(d) 520
Answer: (b) 512000
In simple words: To find the cube of 80, you multiply 80 by itself three times: \( 80 \times 80 \times 80 \). First, calculate \( 8^3 = 512 \). Then, since there is one zero in 80, there will be three zeros in its cube, so \( 512 \text{ with three zeros is } 512000 \).

🎯 Exam Tip: When cubing numbers that end in zeros, cube the non-zero part and then add three times the number of zeros from the original number (e.g., one zero in 80 means three zeros in \( 80^3 \)).

Question 8. The cube root of \( \frac {343}{512} \) is
(a) \( \frac {7}{18} \)
(b) \( \frac {7}{8} \)
(c) \( \frac {-7}{8} \)
(d) \( \frac {-7}{18} \)
Answer: (b) \( \frac {7}{8} \)
In simple words: To find the cube root of a fraction, you find the cube root of the top number (numerator) and the cube root of the bottom number (denominator) separately. The cube root of 343 is 7, and the cube root of 512 is 8. So, the cube root of \( \frac{343}{512} \) is \( \frac{7}{8} \).

🎯 Exam Tip: Memorizing cubes of common numbers (like 7 and 8) helps solve problems involving fractions quickly.

Question 9. The cube-root of 1000 is
(a) 1
(b) 10
(c) 100
Answer: (b) 10
In simple words: We are looking for a number that, when multiplied by itself three times, gives 1000. We know that \( 10 \times 10 \times 10 = 1000 \). So, the cube root of 1000 is 10.

🎯 Exam Tip: For numbers ending in zeros, the cube root will have one-third the number of zeros. Since 1000 has three zeros, its cube root will have one zero.

II. Fill In The Blanks

Question 1. Cubes of even numbers are _______
Answer: even
In simple words: If you take any even number and multiply it by itself three times, the answer will always be an even number. For example, \( 2^3 = 8 \) (even), \( 4^3 = 64 \) (even).

🎯 Exam Tip: Remember that the product of even numbers is always even. Since a cube is an even number multiplied by itself three times, the result must be even.

Question 2. The unit's digit of the cube of 18 is _______
Answer: 2
In simple words: To find the unit's digit of the cube of 18, we only need to look at the unit's digit of 18, which is 8. Then, we find the cube of 8, which is \( 8 \times 8 \times 8 = 64 \times 8 = 512 \). The unit's digit of 512 is 2. The unit digit of a number's cube depends only on the unit digit of the number itself.

🎯 Exam Tip: To quickly find the unit digit of a cube, only cube the unit digit of the original number. For example, for \( 18^3 \), calculate \( 8^3 \), and its unit digit is the answer.

Question 3. The operation of finding the _ is the opposite operation of finding the cube.
Answer: cube root
In simple words: Just like subtraction is the opposite of addition, and division is the opposite of multiplication, finding the cube root is the reverse of finding the cube. If you cube a number, the cube root will bring you back to the original number.

🎯 Exam Tip: Understand inverse operations. For every mathematical operation, there's usually an inverse that undoes it, such as squaring and taking the square root, or cubing and taking the cube root.

Question 4. The cube root is denoted by the _______ sign.
Answer: \( \sqrt[3]{} \)
In simple words: The special symbol for cube root looks like a square root symbol, but it has a small 3 placed above the check mark, inside the left arm. This number 3 tells us we are looking for the cube root, not the square root.

🎯 Exam Tip: Always include the small '3' for the cube root symbol \( (\sqrt[3]{}) \). If the '3' is missing, it implies a square root \( (\sqrt{}) \).

Question 5. The cube root of 8,57,375 is _______
Answer: 95
In simple words: We need to find the number that, when multiplied by itself three times, gives 8,57,375. Since the number ends in 5, its cube root must also end in 5. By checking numbers ending in 5, we find that \( 95 \times 95 \times 95 = 857375 \).

🎯 Exam Tip: To find cube roots of large numbers, first determine the unit digit of the root (e.g., if the number ends in 5, the root ends in 5). Then, separate the last three digits and estimate the tens digit from the remaining number by checking cubes of tens (e.g., \( 90^3 = 729000 \), \( 100^3 = 1000000 \)).

IV. Very Short Answer Type Questions

Question. State True or False for the following statements:
(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.
Answer:
(iii) False. For example, \( 15^2 = 225 \), and \( 15^3 = 3375 \), which ends in 75, not 25. The rule does not hold true in all cases. The last two digits of \( n^3 \) depend on the last two digits of n.
(iv) False. Numbers like 2 and 12, when cubed, end in 8. For instance, \( 2^3 = 8 \) and \( 12^3 = 1728 \). This shows that there are perfect cubes that end with the digit 8.
(v) False. The smallest two-digit number is 10, and its cube is \( 10^3 = 1000 \), which is a four-digit number. All two-digit numbers will have cubes with four or more digits. Thus, a two-digit number's cube cannot be a three-digit number.
(vi) False. The largest two-digit number is 99. Its cube is \( 99^3 = 970299 \), which is a six-digit number. Therefore, the cube of any two-digit number cannot have seven or more digits, as it will always be less than \( 100^3 = 1,000,000 \).
(vii) True. For example, \( 1^3 = 1 \) and \( 2^3 = 8 \). Both 1 and 8 are single-digit numbers. This confirms that the cube of a single-digit number can indeed be a single-digit number.
In simple words: We check each statement by trying small numbers or understanding how cubes work. For numbers ending in 5, their cube might not always end in 25. Numbers like 2 and 12 give cubes ending in 8. The smallest two-digit number, 10, cubes to 1000, which has four digits, so a two-digit number's cube can't be three digits. The biggest two-digit number, 99, cubes to a six-digit number, so its cube cannot have seven or more digits. Finally, small single-digit numbers like 1 and 2 also have single-digit cubes.

🎯 Exam Tip: To answer True/False questions about number properties, always think of specific examples or counter-examples. For digits, focus on the last digit of the base number for predicting the last digit of its power.

Question 1. What do you mean by Hardi- Ramanujan number?
Answer: A Hardi-Ramanujan number is a special type of number that can be expressed as the sum of two cubes in two different ways. For example, 1729 is a famous Hardi-Ramanujan number because it can be written as \( 1^3 + 12^3 \) and also as \( 9^3 + 10^3 \). These numbers are considered interesting in number theory due to their unique properties. Other examples include 4104 and 13832.
In simple words: A Hardi-Ramanujan number is one that can be made by adding two cubed numbers together in two completely different ways. The number 1729 is the smallest example.

🎯 Exam Tip: When defining mathematical terms like "Hardi-Ramanujan number", always include a clear definition and one or two examples to illustrate the concept fully.

Question 2. What do you mean by the cube of a number?
Answer: The cube of a number is the result you get when you multiply that number by itself three times. For instance, if you have the number 'x', its cube would be \( x \times x \times x \), which can also be written as \( x^3 \). This operation calculates the volume of a cube with side length 'x', giving a real-world connection to the term. For example, the cube of 3 is \( 3 \times 3 \times 3 = 27 \).
In simple words: The cube of a number means multiplying the number by itself three times. It shows how big a cube would be if that number was its side.

🎯 Exam Tip: Clearly state that "cube of a number" involves multiplying it three times. Providing an example like \( 3^3=27 \) helps explain the concept perfectly.

Question 3. Express \( 9^3 \) as the sum of consecutive odd numbers?
Answer: To express \( 9^3 \) as the sum of consecutive odd numbers, we first calculate \( 9^3 \).
\( 9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729 \).
The formula for the sum of 'n' consecutive odd numbers that equals \( n^3 \) starts with the odd number \( n^2 - n + 1 \).
For \( n=9 \), the first odd number is \( 9^2 - 9 + 1 = 81 - 9 + 1 = 73 \).
We then list 9 consecutive odd numbers starting from 73:
\( 73 + 75 + 77 + 79 + 81 + 83 + 85 + 87 + 89 = 729 \).
This shows how \( 9^3 \) can be represented as the sum of these nine odd numbers.
In simple words: First, \( 9 \times 9 \times 9 \) is 729. To write 729 as a sum of odd numbers, we start with 73 and add the next 8 odd numbers (75, 77, 79, 81, 83, 85, 87, 89). Adding all these 9 numbers together gives us 729.

🎯 Exam Tip: Remember that \( n^3 \) is the sum of n consecutive odd numbers, and the first odd number in this series is \( n^2 - n + 1 \). This formula simplifies finding the correct starting point.

V. Short Answer Type Questions

Question 1. Suman makes a cuboid of soil of sides 15 cm, 30 cm and 15 cm. How many such cuboids will he need to form a cube?
Answer: The dimensions of Suman's cuboid are 15 cm, 30 cm, and 15 cm.
First, we find the prime factors of each dimension:
\( 15 = 3 \times 5 \)
\( 30 = 2 \times 3 \times 5 \)
Now, we write the combined prime factors of the volume:
Volume factors \( = (3 \times 5) \times (2 \times 3 \times 5) \times (3 \times 5) = 2^1 \times 3^3 \times 5^3 \).
For a number to be a perfect cube, all its prime factors must appear in groups of three. Here, the factor 2 appears only once (as \( 2^1 \)). To make it \( 2^3 \), we need two more factors of 2.
So, we need to multiply by \( 2 \times 2 = 4 \).
Therefore, Suman will need 4 such cuboids to form a perfect cube.
This is because we need \( 2^2 \) more '2's to make the total \( 2^3 \).
In simple words: Suman's cuboid has sides 15, 30, and 15 cm. When we break these numbers into prime factors, we see that '2' appears only once, while '3' and '5' appear three times each. To make a big cube, all prime factors must be in groups of three. Since '2' is only one, we need two more '2's. So, we need to multiply by \( 2 \times 2 = 4 \). This means Suman needs 4 cuboids.

🎯 Exam Tip: To find the number of cuboids needed to form a cube, find the prime factorization of each dimension. Then, ensure each prime factor in the combined volume appears in groups of three by identifying and supplying the missing factors. The product of these missing factors is the number of cuboids required.

Question 2. Mansi has a cuboidal box whose sides are 5 cm, 3 cm and 5 cm. How many such cuboidal boxes will be required for making one cubical box?
Answer: The sides of Mansi's cuboidal box are 5 cm, 3 cm, and 5 cm.
The prime factors of the volume are \( 5 \times 3 \times 5 = 3^1 \times 5^2 \).
For a perfect cube, all prime factors must be in groups of three.
Here, 3 appears once (we need two more 3s, i.e., \( 3^2 \)).
And 5 appears twice (we need one more 5, i.e., \( 5^1 \)).
So, to make the volume a perfect cube, we need to multiply the current factors by \( 3^2 \times 5^1 = (3 \times 3) \times 5 = 9 \times 5 = 45 \).
Therefore, Mansi will need 45 such cuboidal boxes to form one large cubical box.
In simple words: Mansi's box has sides 5, 3, and 5 cm. To make a big cube, all its prime factors (3 and 5) must be found in groups of three. We have one '3' and two '5's. We need two more '3's and one more '5'. So, we need to add \( 3 \times 3 \times 5 = 45 \) more boxes.

🎯 Exam Tip: Always perform prime factorization of the given dimensions. Then, for each prime factor, check if its exponent is a multiple of 3. If not, calculate what power is needed to make it a multiple of 3. The product of these needed powers is the total number of items required.

Question 3. State true or false : for any integer m, \( m^2 < m^3 \) Why?
Answer: The statement "for any integer m, \( m^2 < m^3 \)" is **False**.
Here's why, by considering different cases for 'm':
1. If \( m > 1 \) (positive integers greater than 1):
Let \( m = 2 \). Then \( m^2 = 2^2 = 4 \) and \( m^3 = 2^3 = 8 \). Here, \( 4 < 8 \), so \( m^2 < m^3 \) is true.
Let \( m = 3 \). Then \( m^2 = 3^2 = 9 \) and \( m^3 = 3^3 = 27 \). Here, \( 9 < 27 \), so \( m^2 < m^3 \) is true.
2. If \( m = 1 \):
Then \( m^2 = 1^2 = 1 \) and \( m^3 = 1^3 = 1 \). Here, \( m^2 = m^3 \), so \( m^2 < m^3 \) is false.
3. If \( m = 0 \):
Then \( m^2 = 0^2 = 0 \) and \( m^3 = 0^3 = 0 \). Here, \( m^2 = m^3 \), so \( m^2 < m^3 \) is false.
4. If \( m < 0 \) (negative integers):
Let \( m = -1 \). Then \( m^2 = (-1)^2 = 1 \) and \( m^3 = (-1)^3 = -1 \). Here, \( 1 > -1 \), so \( m^2 > m^3 \). Thus \( m^2 < m^3 \) is false.
Let \( m = -2 \). Then \( m^2 = (-2)^2 = 4 \) and \( m^3 = (-2)^3 = -8 \). Here, \( 4 > -8 \), so \( m^2 > m^3 \). Thus \( m^2 < m^3 \) is false.
Since the statement is not true for \( m=1 \), \( m=0 \), or any negative integer, the general statement "for any integer m, \( m^2 < m^3 \)" is false. This shows the importance of checking all possible cases when proving or disproving a general statement in mathematics.
In simple words: The statement is false because it is not always true. If 'm' is 1 or 0, then \( m^2 \) and \( m^3 \) are equal. If 'm' is a negative number, then \( m^2 \) will be positive and \( m^3 \) will be negative, meaning \( m^2 \) is actually bigger than \( m^3 \). The statement only holds true for positive numbers greater than 1.

🎯 Exam Tip: To prove a "for any integer" statement false, you only need one counter-example. For True/False questions involving variables, always test positive, negative, zero, and one to cover all critical cases.

Question 4. Ratio of three number are 2:3:4 and sum of their cubes is 33957. Find the greatest number.
Answer: Let the three numbers be \( 2x \), \( 3x \), and \( 4x \) according to their given ratio.
The sum of their cubes is 33957. So, we can write the equation:
\( (2x)^3 + (3x)^3 + (4x)^3 = 33957 \)
Now, we calculate the cubes:
\( 8x^3 + 27x^3 + 64x^3 = 33957 \)
Combine the terms with \( x^3 \):
\( (8 + 27 + 64)x^3 = 33957 \)
\( 99x^3 = 33957 \)
To find \( x^3 \), divide 33957 by 99:
\( x^3 = \frac{33957}{99} \)
\( x^3 = 343 \)
Now, we find the cube root of 343:
\( x^3 = 7 \times 7 \times 7 = 7^3 \)
\( \implies \) \( x = 7 \)
The greatest number among \( 2x \), \( 3x \), and \( 4x \) is \( 4x \).
Substitute the value of \( x \):
Greatest number \( = 4 \times 7 = 28 \).
Thus, the greatest of the three numbers is 28.
In simple words: We assume the numbers are 2x, 3x, and 4x. Adding their cubes gives 33957. After solving the equation, we find that x is 7. Since the greatest number is 4x, we multiply 4 by 7 to get 28.

🎯 Exam Tip: When dealing with ratios of numbers and their powers, always represent the numbers as multiples of a common variable (e.g., 2x, 3x, 4x) to maintain the ratio while allowing for algebraic solution.

Question 5. Volume of a cube is 9261000 m³. Find the side of cube.
Answer: Let 'a' be the side of the cube.
The volume of a cube is given by the formula \( V = a^3 \).
We are given that the volume \( V = 9261000 \) m³.
So, \( a^3 = 9261000 \).
To find the side 'a', we need to calculate the cube root of 9261000:
\( a = \sqrt[3]{9261000} \)
We can simplify the number by factoring:
\( 9261000 = 9261 \times 1000 \)
We know that \( \sqrt[3]{1000} = 10 \).
For 9261, we can find its prime factors:
\( 9261 = 3 \times 3087 \)
\( = 3 \times 3 \times 1029 \)
\( = 3 \times 3 \times 3 \times 343 \)
\( = 3^3 \times 7^3 \)
\( = (3 \times 7)^3 \)
\( = 21^3 \)
So, \( \sqrt[3]{9261} = 21 \).
Now, substitute these values back into the equation for 'a':
\( a = \sqrt[3]{21^3 \times 10^3} \)
\( a = 21 \times 10 \)
\( a = 210 \)
Therefore, the side of the cube is 210 meters.
This problem demonstrates finding the cube root of a large number by breaking it into factors of perfect cubes.
In simple words: To find the side of a cube when you know its volume, you need to find the cube root of the volume. We take the cube root of 9261000. By splitting 9261000 into 9261 and 1000, we find the cube root of 9261 is 21, and the cube root of 1000 is 10. Multiplying these gives 210. So, each side of the cube is 210 meters long.

🎯 Exam Tip: When finding the cube root of large numbers, especially those ending in zeros, first separate the number into two factors: the non-zero part and a power of 10 (e.g., \( 9261000 = 9261 \times 1000 \)). Then find the cube root of each factor and multiply the results.