a+b Whole Cube Formula, Proof, and Examples

a+b Whole Cube Formula: Here we shall discuss the algebraic formula of (a+b)³ with its proof derivative and some examples. This is one of the most important and fundamental Algebra formulas that have more significance in competitive exams like NTSE, NDA, AFCAT, SSC, Railways, etc. It is also helpful for school students, especially in class 10th for making mathematical calculations faster.

a + b Whole Cube Formula 

Suppose a and b are two variables which represent 2 algebraic terms. When you add both terms then it will be expressed as a+b in Algebra. It doesn’t tell you only an algebraic expression but also a binomial algebraic equation. The cube of the addition of these algebraic terms a and b is expressed as (a+b)³ with the mathematical formula mentioned below.

a plus b whole cube (a+b)³ is equivalent to the cube of a plus the cube of b plus 3 times the product of a, b and the sum of a plus b. 

The a+b whole cube (a+b)³ algebraic formula is generally applied to calculate the cube of a binomial equation in which two algebraic terms are given. It is also used to make the factors of some types of trinomial equations in which three algebraic terms are given. This a+b whole cube formula is commonly called the cube of the sum of two terms rule, the cube of a binomial identity, and the special binomial product formula.

a + b Whole Cube Formula Proof

For calculating the cube of a binomial equation, we need to multiply three times the algebraic expressions having two terms like (a + b)(a + b)(a + b) = (a + b)³. The formula of a+b whole cube (a + b)³ is also an algebraic identity in mathematics. The (a + b)³ formula is derived as,

(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 

Hence a+b whole cube (a + b)³ formula is finally derived as,

(a + b)³ = a³ + 3a²b + 3ab² + b³

a + b Whole Cube Formula with Examples 

By memorizing this a+b whole cube formula, you can quickly find the solution to binomial algebraic terms in mathematics. Here some examples are given below for a better understanding of this formula a+b whole cube.

Question 1: Find the following algebraic expression with the help of a suitable algebraic identity or formula: (3x + 2y)³

Solution: Given that the expression (3x + 2y)³ has two algebraic terms so it is a binomial equation. 

By using (a + b)³ Algebraic 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: Solve the value of expression 8x³ + y³ when 2x + y = 6 and xy = 2.  

Solution: Given that the expression (8x³ + y³) has two algebraic terms so it is a binomial equation. 

As per the question, 2x + y = 6, and xy = 2

By using (a + b)³ Algebraic Formula,

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here a = 2x; b = y

So, 

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

(2x + y)³ = 8x³ + 12x²y + 6xy² + y³

Putting the value of 2x + y = 6, we get

63 = 8x³ + 6xy(2x + y) + y³

Putting the value of xy = 2, we get

216 = 8x³ + 6 × 2 × 6 + y³

8x³ + y³ = 144

Question 3: Find the value of the given expression with the help of the (a + b)³ formula: (2x + 5y)³

Solution: With the help of (a + b)³ Formula,

(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: Find the value of the given expression with the help of the (a + b)³ formula: (7x + 11y)³

Solution: With the help of (a + b)³ Formula,

(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 algebraic expression by using (a + b)³ identity: (5x + 9y)³

Solution: With the help of (a + b)³ Formula,

(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