What is the Identity a³+b³+c³-3abc? | Simple Explanation for Class 6

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What is the Identity a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca)?

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The identity a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca) is a special algebraic formula that helps us simplify expressions involving cubes of three variables. It shows that if you add the cubes of three numbers and subtract three times their product, it equals the product of their sum and a more complex expression. This identity is very useful for solving complex problems faster.

Simple Example

Quick Example

Imagine you have three friends, A, B, and C, and each wants to calculate a special 'power score'. If A's score is 1, B's is 2, and C's is 3, this identity helps us relate the sum of their 'cube scores' (1³+2³+3³) minus 3 times their product (3*1*2*3) to a simpler multiplication. It's like finding a shortcut to a long calculation!

Worked Example

Step-by-Step

Let's use the identity for a=1, b=2, c=3.
1. First, calculate the Left Hand Side (LHS): a³+b³+c³-3abc
---2. Substitute the values: 1³+2³+3³ - 3(1)(2)(3)
---3. Calculate the cubes and product: 1 + 8 + 27 - 18
---4. Add and subtract: 36 - 18 = 18
---5. Now, calculate the Right Hand Side (RHS): (a+b+c)(a²+b²+c²-ab-bc-ca)
---6. Substitute the values: (1+2+3)(1²+2²+3² - (1*2) - (2*3) - (3*1))
---7. Calculate sums and squares: (6)(1+4+9 - 2 - 6 - 3)
---8. Simplify inside the second bracket: (6)(14 - 11) = (6)(3) = 18
So, LHS = RHS = 18. The identity holds true!

Why It Matters

This identity is a foundational tool in higher mathematics and computer science. Engineers use similar algebraic principles to design efficient circuits, and data scientists apply them to simplify complex data models. Understanding identities like this can open doors to careers in AI/ML, cryptography, and even game development, where optimizing calculations is key.

Common Mistakes

MISTAKE: Forgetting the '-3abc' term on the LHS or the '-ab-bc-ca' terms on the RHS. | CORRECTION: Always remember all parts of the identity. It's a complete formula, not just a partial one.

MISTAKE: Confusing (a+b+c)(a²+b²+c²-ab-bc-ca) with (a+b+c)(a²+b²+c²+ab+bc+ca). | CORRECTION: Notice the MINUS signs before ab, bc, and ca in the correct identity. The plus signs would lead to a different result.

MISTAKE: Incorrectly calculating squares or products inside the brackets, leading to arithmetic errors. | CORRECTION: Always double-check your basic arithmetic, especially squaring numbers and multiplying them.

Practice Questions

Try It Yourself

QUESTION: If a=0, b=1, c=2, what is the value of a³+b³+c³-3abc? | ANSWER: 9

QUESTION: Using the identity, find the value of 2³+3³+ (-5)³ - 3(2)(3)(-5). | ANSWER: 0

QUESTION: If a+b+c=0, what can you say about a³+b³+c³-3abc? (Hint: Look at the RHS of the identity) | ANSWER: If a+b+c=0, then a³+b³+c³-3abc = 0, which means a³+b³+c³ = 3abc.

MCQ

Quick Quiz

Which of the following is equal to a³+b³+c³-3abc?

(a+b+c)(a²+b²+c²+ab+bc+ca)

(a+b+c)(a²+b²+c²-ab-bc-ca)

(a-b-c)(a²+b²+c²-ab-bc-ca)

(a+b+c)(a²+b²+c²)

The Correct Answer Is:

Option B correctly represents the identity with the sum of terms (a+b+c) multiplied by the expression (a²+b²+c²-ab-bc-ca). Options A, C, and D have incorrect signs or missing terms.

Real World Connection

In the Real World

Imagine a civil engineer designing a bridge. They use complex equations to calculate forces and stresses. Sometimes, these equations involve terms like a³, b³, c³. By using algebraic identities like this, they can simplify calculations, making the design process faster and more accurate. This helps in building safer structures, just like how ISRO scientists use advanced math to calculate rocket trajectories!

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of the variables | CUBE: A number multiplied by itself three times (e.g., a³ = a*a*a) | EXPRESSION: A combination of numbers, variables, and operation symbols | VARIABLE: A symbol (like a, b, c) that represents a quantity that can change | SIMPLIFY: To reduce an expression to a simpler form

What's Next

What to Learn Next

Great job understanding this identity! Next, you can explore other important algebraic identities like (a+b)³ or (a-b)³. These will build on your knowledge and help you tackle even more complex algebraic problems in higher classes.