(a + b)³ Formula: Cube of a Sum Explained with Examples

(a + b)³ Formula: In this guide, we provide a detailed explanation of the (a + b)³ algebraic formula, including its mathematical derivation and practical examples. This identity is one of the essential Algebra formulas frequently tested in competitive exams such as NTSE, NDA, AFCAT, SSC, and Railways. It is a vital tool for class 10th students to simplify complex expressions and accelerate mathematical calculations.

(a + b)³ Formula 

Let a and b represent two distinct algebraic terms. When added, they form the expression a+b, which is a fundamental binomial algebraic equation. The cube of this sum is written as (a+b)³, and its expanded form is defined by the following mathematical identity.

(a + b)³ = a³ + b³ + 3ab(a + b)

The a plus b whole cube (a + b)³ identity is defined as the cube of 'a' plus the cube of 'b' plus 3 times the product of 'a' and 'b', multiplied by the sum (a + b). 

(a + b)³ = a³ + b³ + 3a²b + 3ab²

The a+b whole cube (a + b)³ formula is primarily used to solve the cube of binomial expressions. It also serves as a critical method for factoring trinomials. This identity is widely recognized as the cube of a sum rule, a standard binomial identity, and a special product formula in algebra.

a + b Whole Cube Formula Proof

To compute the cube of a binomial, you must multiply the term by itself three times: (a + b)(a + b)(a + b) = (a + b)³. Because the a+b whole cube formula represents a standard algebraic identity, we can derive it systematically as shown below:

(a + b)³ = (a + b)(a + b)(a + b)

= (a² + 2ab + b²)(a + b)

= a³ + a²b + 2a²b + 2ab² + ab² + b³ 

= a³ + 3a²b + 3ab² + b³ 

= a³ + 3ab(a+b) + b³ 

Therefore, the a+b whole cube (a + b)³ formula is derived as follows:

(a + b)³ = a³ + 3a²b + 3ab² + b³

a + b Whole Cube Formula with Examples 

Memorizing this formula allows you to solve binomial problems efficiently. Below are several worked examples to help you master the application of this algebraic identity.

Question 1: Expand the following expression using the appropriate algebraic identity: (3x + 2y)³

Solution: Given the expression (3x + 2y)³, we recognize it as a binomial equation ready for the expansion formula. 

Applying the (a + b)³ formula:

(a + b)³ = a³ + 3a²b + 3ab² + b³

(3x + 2y)³ = (3x)³ + 3 × (3x)² × 2y + 3 × (3x) × (2y)² + (2y)³

(3x + 2y)³ = 27x³ + 54x²y + 36xy² + 8y³

Question 2: Calculate the value of 8x³ + y³ given that 2x + y = 6 and xy = 2.  

Solution: We use the binomial expansion identity to relate the given values. 

Given: 2x + y = 6, and xy = 2

Using the (a + b)³ formula:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Let a = 2x and b = y

Substituting these values: 

(2x + y)³ = (2x)³ + 3 × (2x)² × (y) + 3 × 2x × (y)² + (y)³

(2x + y)³ = 8x³ + 12x²y + 6xy² + y³

Substituting 2x + y = 6, we get:

6³ = 8x³ + 6xy(2x + y) + y³

Using xy = 2, we get:

216 = 8x³ + 6 × 2 × 6 + y³

8x³ + y³ = 144

Question 3: Expand the following expression using the (a + b)³ formula: (2x + 5y)³

Solution: Applying the standard (a + b)³ expansion:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here, a = 2x and b = 5y

(2x + 5y)³ = (2x)³ + 3 × (2x)² × 5y + 3 × (2x) × (5y)² + (5y)³

(2x + 5y)³  = 8x³ + 60x²y + 150xy² + 125y³

Question 4: Expand the following expression using the (a + b)³ formula: (7x + 11y)³

Solution: Applying the standard (a + b)³ expansion:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here, a = 7x and b = 11y

(7x + 11y)³ = (7x)³ + 3 × (7x)² × 11y + 3 × (7x) × (11y)² + (11y)³

(7x + 11y)³  = 343x³ + 1617x²y + 2541xy² + 1331y³

Question 5: Calculate the value of the following expression using the (a + b)³ identity: (5x + 9y)³

Solution: Applying the standard (a + b)³ expansion:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here, a = 5x and b = 9y

(5x + 9y)³ = (5x)³ + 3 × (5x)² × 9y + 3 × (5x) × (9y)² + (9y)³

(5x + 9y)³  = 125x³ + 675x²y + 405xy² + 729y³