NCERT Solutions for Class 8 Maths Chapter 6 Cubes and Cube Roots

NCERT Solutions for Class 8 Maths Chapter 6 – Cubes and Cube Roots are helpful tools for students to learn how to find the cube of a number and its root. These solutions help you understand how to multiply a number three times by itself. By practicing these cubes and cube roots class 8 questions and answers, students can master the methods of prime factorization and estimation to solve problems quickly.

Understanding Cubes and Cube Roots

To start with, we should know what a cube is in math. When we multiply a number by itself three times, we get its cube. For example, if we take the number 2 and multiply it like 2 × 2 × 2, we get 8. So, 8 is the cube of 2. On the other hand, the cube root is the opposite. It tells us which number was multiplied three times to get that big number.

What are Perfect Cubes?

A perfect cube is a number that comes from multiplying a whole number by itself three times.

• 1 × 1 × 1 = 1

• 2 × 2 × 2 = 8

• 3 × 3 × 3 = 27

• 4 × 4 × 4 = 64

We can check if a number is a perfect cube using prime factorisation. If every prime factor appears in a group of three, it is a perfect cube. If any factor is left alone or in a group of two, it is not a perfect cube. This is a vital part of cubes and cube roots class 8 questions and answers.

Interesting Cube Patterns

There are many fun patterns in cubes. For instance, the cube of an even number is always even. The cube of an odd number is always odd. Also, look at the ending digits. If a number ends in 1, its cube also ends in 1. If it ends in 4, the cube ends in 4. These small tips help you solve cubes and cube roots class 8 NCERT solutions much faster during your exams.

Finding Roots Using Prime Factors

The prime factorisation method is the best way to find cube roots for school exams. We break the number down into its smallest prime parts. After that, we group those parts into sets of three. From each set, we take one number and multiply them together to get our final answer.

Steps for Prime Factorisation

1. Divide the number by the smallest prime number like 2, 3, or 5.

2. Keep dividing until you reach 1.

3. Write all the factors together.

4. Make groups of three same factors.

5. Multiply one number from each group.

This method is used in many cubes and cube roots class 8 extra questions with answers. It ensures you don't make mistakes with big numbers. 

Smallest Multiple for Cubes

Sometimes, a number is not a perfect cube. In your cubes and cube roots class 8 questions and answers, you might be asked to find a number to multiply or divide.

• To Multiply: Find the factors that are not in a group of three. Add more of that factor to make a group of three.

• To Divide: Find the extra factors that don't fit in a group of three. Divide the main number by these extras to get a perfect cube.

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Estimating the Cube Roots Quickly

Did you know you can guess a cube root without doing long math? This is called the estimation method. It is very handy for short questions in NCERT Solutions for Class 8 Maths Chapter 6 – Cubes and Cube Roots. You first group the digits of the number into threes, starting from the right side.

How Estimation Works

• Group 1: The last three digits tell us the last digit of the cube root.

• Group 2: The remaining digits tell us the first digit of the cube root.

• For example, in 1331, the last group is 331. Since it ends in 1, the root's last digit is 1. The first group is 1, so the first digit is 1. The answer is 11.

This trick is perfect for solving cubes and cube roots class 8 extra questions with answers quickly. It saves time and helps you check if your prime factorisation answer is correct. Students love this method because it feels like a math magic trick.

Harday-Ramanujan Numbers

In this chapter, we also learn about special numbers like 1729. These are called Hardy-Ramanujan numbers. They are special because they can be written as the sum of two cubes in two different ways. For example:

• $1729 = 1728 + 1 = 12^3 + 1^3$

• $1729 = 1000 + 729 = 10^3 + 9^3$
  Learning these facts makes your cubes and cube roots class 8 NCERT solutions study session more fun and interesting.

Solved Examples for Better Practice

Practice is the key to doing well in math. We have picked some common problems found in cubes and cube roots class 8 pdf answers questions and guides. These will help you understand how to write answers in your school notebook.

Example 1: Is 216 a perfect cube?

• Step 1: Prime factors of 216 are 2 × 2 × 2 × 3 × 3 × 3.

• Step 2: Group them into threes: (2 × 2 × 2) and (3 × 3 × 3).

• Step 3: Since all factors are in groups of three, 216 is a perfect cube.

• Step 4: Cube root = 2 × 3 = 6.

Example 2: Smallest number to multiply with 72

• Step 1: Factors of 72 are 2 × 2 × 2 × 3 × 3.

• Step 2: Here, 2 is in a group of three, but 3 is only there twice.

• Step 3: We need one more 3 to make a triplet.

• Result: Multiply 72 by 3 to get 216, which is a perfect cube.

Using these cubes and cube roots class 8 questions and answers will make you feel confident. Always remember to show every step clearly so your teacher can give you full marks.

Tips for Solving Math Easily

Math doesn't have to be hard. If you follow a plan, you can solve any problem. Here are some simple tips for your NCERT Solutions for Class 8 Maths Chapter 6 Cubes and Cube Roots study time. Don't rush through the numbers; take your time to group them correctly.

Daily Study Habits

When you solve cubes and cube roots class 8 extra questions with answers, try to do it without looking at the solution first. This will help your brain think harder.

Summary of Rules

• The cube of a negative number is always negative.

• A perfect cube cannot end with two zeros (it must have three, six, or nine zeros).

• If a number ends in 5, its cube also ends in 5.

• The cube root of a number ending in 8 will always end in 2.

These rules are like shortcuts. They help you find cubes and cube roots class 8 questions and answers without doing the whole math on paper. Keep practicing every day, and you will find this chapter very easy!

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