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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.
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
Related Links:
| a2 b2 Formula | Trigonometry Formulas |
| Standard deviation formula | Maths Formulas |
Frequently Asked Questions (FAQs) on (a + b) Whole Cube Formula
The formula for (a + b) whole cube is (a + b)³ = a³ + 3a²b + 3ab² + b³.
The formula for (a - b) whole cube is (a - b)³ = a³ - 3a²b + 3ab² - b³.
The formula for (a + b) whole square is (a + b)² = a² + 2ab + b².
The identity of (a + b)³ is a³ + 3a²b + 3ab² + b³.
The formula for a³ + b³ is (a + b)(a² - ab + b²). What is the formula for (a + b) whole cube?
What is the formula for (a - b) whole cube?
What is the formula for (a + b) whole square?
What is the identity of (a + b)³?
What is the formula for a³ + b³?
