(a + b) Whole Cube Formula

# (a + b) Whole Cube Formula

The algebraic formula of (a + b) Whole Cube is a fundamental and powerful mathematical expression widely used in various competitive exams and academic settings. The (a + b) Whole Cube Formula allows us to efficiently expand the cube of a binomial expression (a + b) and simplifies calculations involving polynomial expressions. Understanding the (a + b) Whole Cube Formula is crucial for students, particularly those in class 10th, as it enables them to expedite mathematical computations and solve problems with ease.

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In this discussion, we will delve into the derivation of the (a + b) Whole Cube formula, explore its applications, and provide illustrative examples to demonstrate its practical utility. Mastering the (a + b) Whole Cube formula will enhance problem-solving skills and lay a strong foundation for tackling more advanced mathematical concepts in the future.

## (a + b) Whole Cube Formula

In algebra, the cube of the sum of two algebraic terms, a and b, is represented as (a + b)³. This powerful formula is essential in expanding and simplifying polynomial expressions. To compute (a + b)³, we break it down into three components: the cube of ‘a,’ the cube of ‘b,’ and the product of 3ab multiplied by the sum of ‘a’ and ‘b.’
The expression can be written as:

(a + b)3 = a³ + b³ + 3ab(a + b)
(a + b)3 = a³ + b³ + 3a²b + 3ab²

### (a + b) Whole Cube Formula Derivation/Proof

To derive the formula for (a + b)³, we start with the multiplication of three algebraic expressions of the binomial (a + b)(a + b)(a + b) = (a + b)³. Then, we simplify the expression step by step as follows:

Step 1: Multiply the first two binomials using the distributive property.

(a + b)(a + b) = a² + 2ab + b²

Step 2: Now, multiply the result of step 1 with the third binomial (a + b).

(a² + 2ab + b²)(a + b) = a³ + a²b + 2a²b + 2ab² + ab² + b³

Step 3: Combine like terms in the expression obtained in step 2.

a³ + (a²b + 2a²b) + (2ab² + ab²) + b³ = a³ + 3a²b + 3ab² + b³

Step 4: Notice that the terms (a²b + 2a²b) and (2ab² + ab²) can be simplified to 3a²b and 3ab², respectively.

Step 5: Factor out 3ab from the last three terms.
a³ + 3ab(a + b) + b³

Thus, the final result is the formula for (a + b)³:

(a + b)3 = a³ + 3a²b + 3ab² + b³

This (a + b) Whole Cube formula is essential in expanding the cube of a binomial and simplifying expressions in algebraic equations, making mathematical calculations more efficient and straightforward.

### (a + b) Whole Cube Formula with Examples

The examples below demonstrate the application of the (a + b) Whole Cube Formula to calculate the cube of various binomial expressions. By understanding and memorizing the (a + b) Whole Cube formula, one can quickly and efficiently solve such algebraic problems, making mathematical computations faster and more convenient.

Example 1: Find the cube of (2x + 3y).

Solution:
Using the (a + b) Whole Cube formula:
(2x + 3y)³ = (2x)³ + 3(2x)²(3y) + 3(2x)(3y)² + (3y)³
= 8x³ + 36x²y + 54xy² + 27y³

Example 2: Calculate the cube of (a – 5b).

Solution:
Using the (a + b) Whole Cube formula:
(a – 5b)³ = (a)³ + 3(a)²(-5b) + 3(a)(-5b)² + (-5b)³
= a³ – 15a²b + 75ab² – 125b³

Example 3: Determine the cube of (3x² – 4y).

Solution:
Using the (a + b) Whole Cube formula:
(3x² – 4y)³ = (3x²)³ + 3(3x²)²(-4y) + 3(3x²)(-4y)² + (-4y)³
= 27x^6 – 108x^4y + 144x²y² – 64y³

## Frequently Asked Questions (FAQs) on (a + b) Whole Cube Formula

### What is the formula for (a + b) whole cube?

The formula for (a + b) whole cube is (a + b)³ = a³ + 3a²b + 3ab² + b³.

### What is the formula for (a - b) whole cube?

The formula for (a - b) whole cube is (a - b)³ = a³ - 3a²b + 3ab² - b³.

### What is the formula for (a + b) whole square?

The formula for (a + b) whole square is (a + b)² = a² + 2ab + b².

### What is the identity of (a + b)³?

The identity of (a + b)³ is a³ + 3a²b + 3ab² + b³.

### What is the formula for a³ + b³?

The formula for a³ + b³ is (a + b)(a² - ab + b²).

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