What is (a+b) Whole Cube?

S3-SA1-0045

Grade Level:

Class 6

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

Definition

What is it?

The expression (a+b) whole cube, written as (a+b)^3, means multiplying the sum (a+b) by itself three times. It's a special algebraic identity that helps us quickly expand expressions like (2+3)^3 without doing long multiplication. This identity simplifies calculations involving cubes of binomials.

Simple Example

Quick Example

Imagine you have two friends, Rahul and Priya, and you want to calculate the total number of ladoos they get if each person's share is 'a' and 'b' respectively, and this whole combined share is cubed. If Rahul gets 2 ladoos (a=2) and Priya gets 3 ladoos (b=3), then (2+3)^3 means (5)^3, which is 5 x 5 x 5 = 125 ladoos. The (a+b)^3 formula helps us find this total in a structured way.

Worked Example

Step-by-Step

Let's expand (x+2)^3 using the identity (a+b)^3 = a^3 + b^3 + 3ab(a+b).

Step 1: Identify 'a' and 'b'. Here, a = x and b = 2.
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Step 2: Substitute 'a' and 'b' into the formula: x^3 + 2^3 + 3(x)(2)(x+2).
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Step 3: Calculate the cube of 'b'. 2^3 = 2 x 2 x 2 = 8.
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Step 4: Multiply the terms in 3ab(a+b). 3(x)(2) = 6x. So, we have 6x(x+2).
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Step 5: Distribute 6x into (x+2). 6x * x = 6x^2 and 6x * 2 = 12x.
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Step 6: Combine all the terms: x^3 + 8 + 6x^2 + 12x.
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Step 7: Arrange the terms in decreasing order of powers (standard form): x^3 + 6x^2 + 12x + 8.

Answer: (x+2)^3 = x^3 + 6x^2 + 12x + 8.

Why It Matters

Understanding (a+b) whole cube is fundamental for solving complex problems in various fields. Engineers use it to design structures and calculate volumes, while data scientists might use similar polynomial expansions to model data trends. Cryptographers even use these algebraic principles to create secure online transactions, protecting your digital payments.

Common Mistakes

MISTAKE: Students often write (a+b)^3 as a^3 + b^3. | CORRECTION: Remember that (a+b)^3 is not just the sum of cubes; it's (a+b) multiplied by itself three times, which includes extra terms like 3a^2b and 3ab^2.

MISTAKE: Forgetting to distribute the 3ab term correctly, writing 3ab(a+b) as 3a^2b + 3ab. | CORRECTION: Always remember to multiply 3ab by BOTH 'a' and 'b' inside the bracket, resulting in 3a^2b + 3ab^2.

MISTAKE: Making calculation errors with the exponents, especially when 'a' or 'b' are numbers. For example, writing 2^3 as 6 instead of 8. | CORRECTION: Double-check all multiplications for powers. 2^3 means 2 x 2 x 2, not 2 x 3.

Practice Questions

Try It Yourself

QUESTION: Expand (y+3)^3. | ANSWER: y^3 + 9y^2 + 27y + 27

QUESTION: If (a+b)^3 = a^3 + b^3 + P, what is P? | ANSWER: P = 3ab(a+b) or 3a^2b + 3ab^2

QUESTION: Find the value of (10+1)^3 using the identity. | ANSWER: (10+1)^3 = 10^3 + 1^3 + 3(10)(1)(10+1) = 1000 + 1 + 30(11) = 1001 + 330 = 1331

MCQ

Quick Quiz

Which of the following is the correct expansion of (x+y)^3?

x^3 + y^3

x^3 + y^3 + 3xy(x+y)

x^3 + y^3 + 3xy

x^3 + y^3 + x^2y + xy^2

The Correct Answer Is:

Option B is the correct algebraic identity for (x+y)^3. Options A, C, and D are incomplete or incorrect expansions, missing key terms from the identity.

Real World Connection

In the Real World

Imagine an architect designing a new building in Mumbai. If a room has sides (length + width) and this combined dimension needs to be cubed for a special artistic feature or structural calculation, they would use (a+b)^3. It helps them quickly find the volume or other properties without tedious individual calculations, making their design process faster and more accurate.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of its variables. | BINOMIAL: An algebraic expression with two terms, like (a+b). | EXPANSION: The process of multiplying out terms in an expression to remove brackets. | CUBE: Multiplying a number or expression by itself three times (e.g., x^3).

What's Next

What to Learn Next

Great job understanding (a+b) whole cube! Next, you should explore 'What is (a-b) Whole Cube?'. It's very similar but with some important sign changes, which will strengthen your grasp of algebraic identities even further.