Class 8 Maths: Cube and Cube Roots Quiz

Class 8 Maths: Understanding Cube and Cube Roots

In Class 8 Maths, students are introduced to new concepts and skills that help them better understand and solve problems related to mathematics. One of the key topics in this class is the study of cube and cube roots. In this article, we will delve into the world of cubes and cube roots, exploring their definitions, properties, and how to find cube roots of a number.

What are Cubes and Cube Roots?

A cube is a mathematical figure whose edges are equal and form a perfect cube shape. When the sides of the cube are of length a unit, then the cube describes a cube of volume (a x a x a) = a³ cubic units.

The cube root of a number is the number that, when multiplied by itself three times, gives the original number. In mathematical terms, if a is a real number, a³ is the cube of a, and ³√a is the cube root of a.

Properties of Cubes and Cube Roots

There are several properties associated with cubes and cube roots, including:

1. Associative Property: The cube of the sum or difference of two numbers is equal to the sum or difference of their cubes: (a + b)³ = a³ + 3a²b + 3ab² + b³ and (a - b)³ = a³ - 3a²b - 3ab² + b³.

2. Distributive Property: The cube of the product of two numbers is equal to the product of their cubes: (ab)³ = a³b³.

3. Identity Property for Cube Roots: Every positive number has a unique cube root. If a is a positive number, then there exists a unique number x such that a = x³.

4. Identity Property for Cubes: Every positive number has a unique cube. If a is a positive number, then there exists a unique number x such that a = x³.

5. Reflexive Property: For any number a, a³ = a³.

6. Symmetric Property: For any numbers a and b, a³ = b³ if and only if a = b.

7. Transitive Property: For any numbers a, b, and c, if a³ = b³ and b³ = c³, then a³ = c³.

8. Cube Root Property: For any numbers a and b, if a³ = b³, then a = b.

9. Cube Property: For any numbers a, b, and c, if a³ = bc, then a = √[b³] and a = √[c³].

Finding Cube Roots of a Number

To find the cube root of a number, you can use the following methods:

1. Long Division Method: This method involves dividing the given number by a power of 3 repeatedly until the divisor is 1. Then, the quotient is the cube root of the given number.

2. Cubing Method: This method involves finding three numbers that, when multiplied together, give the given number. Then, the cube root of the given number is the average of these three numbers.

3. Trial and Error Method: This method involves guessing a number and then finding its cube. If the cube is not the given number, adjust the guess and try again until the cube is the given number.

Practice Problems

1. Find the cube of the following numbers: a. 2 b. 3 c. 4 d. 5
2. Find the cube root of the following numbers: a. 3 b. 8 c. 27 d. 125

Conclusion

Cubes and cube roots are essential mathematical concepts that students learn in Class 8 Maths. By understanding the properties of cubes and cube roots and learning how to find cube roots of a number, students can develop the skills necessary to solve more complex mathematical problems. This knowledge will also prepare them for more advanced mathematics courses in the future.