Summary of CBSE Class 8 Mathematics - Cubes and cube roots

Here is an overview of some of the main topics covered under the chapter "Cubes and Cube Roots" in CBSE Class 8 Mathematics:

1. Definition and properties of cubes: Students learn the definition of a cube, its properties, such as the number of faces, edges, and vertices. They also get an understanding of the volume and surface area of cubes.
2. Cube roots: The concept of cube roots is explained, and students learn to determine the cube root of a perfect cube. They learn different methods to find the cube root of a number like prime factorization, estimation, repeated subtraction method, etc.
3. Perfect cubes: Students learn about perfect cubes from 1 to 1000. They memorize them for quick calculations.
4. Sum and difference of cubes: The topic possibly involves the formulas for the cube of the sum or difference of two numbers and then understand how to factorize the expressions by applying the factor theorem or the splitting middle term method.
5. Solving problems using cubes and cube roots: Students solve various problems related to cubes and cube roots by using the concepts covered in the chapter.

Overall, students cover a wide range of topics in "Cubes and Cube Roots" in CBSE Class 8 Mathematics, which helps in building a strong foundation in Mathematics.

Key notes of CBSE Class 8 Mathematics - Cubes and cube roots

1. A cube is a three-dimensional shape with six equal square faces.
2. The area of one face of a cube is given by a² and the total surface area of a cube is given by 6a².
3. The volume of a cube is given by V = a³, where a is the length of one side of the cube.
4. The cube root of a number is the number which when multiplied by itself three times gives the original number.
5. To find the cube root of a number, we can use methods like prime factorization, repeated subtraction, estimation, etc.
6. The sum and difference of cubes identities can be used to expand expressions like (a + b)³ or (a - b)³.
7. To factorize a cubic expression, we can apply the factor theorem or the splitting middle term method.

Practice questions and answers of CBSE Class 8 Mathematics - Cubes and cube roots

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1. What is a perfect cube?

Answer: A perfect cube is a number that can be obtained by multiplying an integer by itself three times.

2.What is the relationship between the volume and the length of a cube?

Answer: The volume of a cube is given by the formula V = a³, where a is the length of one side of the cube. Therefore, if you know the length of one side, you can easily find the volume of the cube.

3. What is the cube root of 729?

Answer: The cube root of 729 is 9, because 9 cubed (9³) is equal to 729.

4. What is the sum of cubes formula?‍

Answer: The sum of cubes formula is (a + b)³ = a³ + 3a²b + 3ab² + b³, where a and b are any two numbers.

5. What is the difference of cubes formula?‍

Answer: The difference of cubes formula is (a - b)³ = a³ - 3a²b + 3ab² - b³, where a and b are any two numbers.

6. How can you find the cube root of a number using prime factorization?

Answer: To find the cube root of a number using prime factorization, you need to prime factorize the given number and then take the cube root of each of the prime factors. Finally, multiply all the cube roots obtained to get the required cube root of the given number.

For example, to find the cube root of 64, we first prime factorize it as 64 = 2³ × 2³. Taking the cube root of each prime factor, we get 2 as the cube root in each case. Finally, multiplying both cube roots, we get the cube root of 64 as 2.

7. How to estimate the cube root of a number?

Answer: To estimate the cube root of a number, we first round off the number to a nearest perfect cube. Then, we find the cube root of that perfect cube. This gives us an estimate of the required cube root.

For example, to estimate the cube root of 200, we round off 200 to the nearest perfect cube, which is 125. The cube root of 125 is 5. Therefore, an estimate of the cube root of 200 is 5.

8. What is the volume of a cube with side length 4 cm?

Answer: The volume of a cube is given by the formula V = a³, where a is the length of one side of the cube. Therefore, if the side length of a cube is 4 cm, its volume is V = 4³ = 64 cubic cm.

9. Can the cube root of a number be negative?‍

Answer: Yes, the cube root of a number can be negative. This is because the cube of a negative number is negative, and the cube of a positive number is positive. Therefore, a negative number multiplied by itself three times gives a negative number, which means the cube root of a negative number is also negative.

10. What is the total surface area of a cube with side length 6 cm?

Answer: The total surface area of a cube is given by the formula A = 6a², where a is the length of one side of the cube. Therefore, if the side length of a cube is 6 cm, its total surface area is A = 6 × 6² = 216 square cm.

11. What is the cube of 5?

Answer: The cube of 5 is 125, because 5 multiplied by itself three times (5³) gives 125.

12. What is the cube root of 27?

Answer: The cube root of 27 is 3, because 3 multiplied by itself three times (3³) gives 27.

13. What is the relationship between the cube of a number and the cube root of its cube?‍

Answer: The cube of a number and the cube root of its cube are inverse operations of each other. For example, if the cube of a number is 64, its cube root is 4, because 4³=64.

14. How can you find the cube root of a decimal number?

Answer: To find the cube root of a decimal number, we can first convert the decimal to a fraction. Then, we can find the cube root of the numerator and the denominator separately, and divide the cube root of the numerator by the cube root of the denominator to get the required cube root of the decimal.

15. Can you simplify the expression (4a³ + 3b³) - (2a³ - 2b³)?

Answer: Yes, we can simplify the expression (4a³ + 3b³) - (2a³ - 2b³) as follows:

(4a³ + 3b³) - (2a³ - 2b³) = 4a³ + 3b³ - 2a³ + 2b³
= 2a³ + 5b³

Therefore, the simplified form of the expression is 2a³ + 5b³.

High difficulty questions and answers of CBSE Class 8 Mathematics - Cubes and cube roots

1. What is the cube of the square root of 5?

Answer: The cube of the square root of 5 is (√5)³ = 5√5.

2. What is the smallest number greater than 126 that is a perfect cube?

Answer: We can find the smallest number greater than 126 that is a perfect cube by taking the cube root of 126, which is approximately 5.02. Rounding this up to the nearest integer yields 6, which is the smallest integer greater than 126 that is a perfect cube. Therefore, the answer is 6³ = 216.

3. What is the difference between the sum of the cubes of the first n natural numbers and the square of their sum?

Answer: The sum of the cubes of the first n natural numbers is given by the formula (1+2+3+...+n)². The square of the sum of the first n natural numbers is (1³+2³+3³+...+n³). Therefore, the difference between the two is (1+2+3+...+n)² - (1³+2³+3³+...+n³).

This expression is difficult to simplify, but it can be written as a summation notation as follows: ∑i =1 to n i² + 2∑i<j<j=i to n i*j - ∑i=1 to n i³.

This expression can be calculated numerically for a given value of n.

4. How many digits are in the cube root of 5?

Answer: The cube root of 5 is an irrational number that cannot be expressed as a finite decimal or a fraction. Therefore, it has an infinite number of digits after the decimal point. However, if we are only interested in the number of digits before the decimal point, we can estimate it by finding the largest integer whose cube is less than 5. Using trial and error, we find that the largest integer whose cube is less than 5 is 1, so the cube root of 5 is between 1 and 2. Therefore, the number of digits before the decimal point is 1.

5. What is the cube root of -8?

Answer: The cube root of -8 is -2, because -2 x -2 x -2 = -8.

6. What is the value of x if 2x + 5 = cube root of 27?

Answer: We can begin by writing the cube root of 27 as the cube root of 3³, which is 3. Therefore, we have 2x + 5 = 3, which simplifies to 2x = -2. Solving for x by dividing both sides by 2 gives x = -1.

7. What is the cube of (a+b) if a = 2 and b = 3?

Answer: The cube of (a+b) can be expanded using the formula (a+b)³ = a³ + 3a²b + 3ab² + b³. Substituting a = 2 and b = 3 gives (2+3)³ = 125.

8. What is the unit digit of the cube of 2019?

Answer: The unit digit of the cube of 2019 depends only on the unit digits of 2019. Since the unit digit of 2019 is 9, the unit digit of its cube is the same as the unit digit of 9³, which is 1. Therefore, the unit digit of the cube of 2019 is 1.

9. What is the cube root of (x³+y³)/3 if x and y are positive real numbers?

Answer: Applying the formula for the sum of cubes, we can write (x³+y³)/3 = [(x+y)/2]³ - 3[(x+y)/2]²(x/2) + 3[(x+y)/2](x/2)² - (y/2)³. Taking the cube root of both sides gives the formula (x+y)/2 - [3(x+y)/2² x/2]/(3x(x+y)/4) + [3(x+y)/2 (x/2)²]/(3x(x+y)/4) - (y/2)/cbrt(27).

Simplifying this expression gives (x+y)/2 - (3xy)/(2(x+y)) + (y/2cbrt(3)) = ((x+y)² - 3xy)/(2cbrt(3)(x+y)).

Therefore, the cube root of (x³+y³)/3 is ((x+y)² - 3xy)/(2cbrt(3)(x+y)).

10. How many positive integers less than 1000 are perfect cubes?

Answer: We can find the largest integer n such that n³ is less than 1000 by taking the cube root of 1000, which is approximately 10. Therefore, the positive integers less than 1000 that are perfect cubes are 1, 8, 27, 64, 125, 216, and 343. This is a total of 7 positive integers.