Mathematics Class 8 - Cubes And-Cube-Roots Notes

Cubes and Cube Roots

Welcome to the chapter on Cubes and Cube Roots for Class 8. In this chapter, you will learn what cubes and cube roots are, how to find them, and how to solve problems using these concepts. By the end of this chapter, you will be able to calculate cubes and cube roots of numbers and use them in word problems!

Introduction

Cubes and cube roots are important concepts in mathematics. A cube is the product of a number multiplied by itself three times. The cube root is the number that, when multiplied by itself three times, gives the original number.

What is a Cube?

The cube of a number is found by multiplying the number by itself twice more.

• Cube of 2: 2 × 2 × 2 = 8
• Cube of 3: 3 × 3 × 3 = 27
• Cube of 5: 5 × 5 × 5 = 125

Properties of Cubes

• The cube of any integer is always a positive integer.
• Cubes of even numbers are even; cubes of odd numbers are odd.
• The last digit of a cube depends on the last digit of the original number.

Perfect Cubes

A perfect cube is a number that can be written as the cube of an integer.

• 8 is a perfect cube (2 × 2 × 2)
• 27 is a perfect cube (3 × 3 × 3)
• 64 is a perfect cube (4 × 4 × 4)

What is a Cube Root?

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. The symbol for cube root is ∛.

• Cube root of 8: ∛8 = 2
• Cube root of 27: ∛27 = 3
• Cube root of 125: ∛125 = 5

Finding Cube Roots

• By prime factorization: Break the number into prime factors and group them in threes.
• By estimation: Find the nearest perfect cubes.

Word Problems

Use cubes and cube roots to solve real-life problems. Read the question, decide if you need to find a cube or cube root, and solve.

• Example: If the volume of a cube is 125 cm³, what is the length of one side? (∛125 = 5 cm)

Fun Activity: Cube Numbers Game!

Write down numbers from 1 to 10 and find their cubes. See who can find the most cube roots of perfect cubes!

Summary

• A cube is the product of a number multiplied by itself three times.
• A cube root is the number that gives the original number when cubed.
• Use prime factorization or estimation to find cube roots.

Practice Questions

1. Find the cube of 6.
2. Is 216 a perfect cube? If yes, what is its cube root?
3. Find the cube root of 64.
4. What is the cube of 9?
5. If the volume of a cube is 343 cm³, what is the length of one side?

Challenge Yourself

• List all perfect cubes between 1 and 1000.
• Find the cube root of 1000 using prime factorization.

Did You Know?

• The cube of a negative number is always negative!
• Cube roots are used in finding the side of a cube when its volume is known.

Glossary

• Cube: The product of a number multiplied by itself three times.
• Cube Root: The number that gives the original number when cubed.
• Perfect Cube: A number that is the cube of an integer.
• Prime Factorization: Breaking a number into its prime factors.

Answers to Practice Questions

1. 216
2. Yes, cube root is 6.
3. 4
4. 729
5. 7 cm

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Practice cubes and cube roots to become confident in solving math problems!