a³ + b³ formula which is also called the sum of cubes formula, in algebra is the fundamental algebra formula that is used to find the sum of cubes of any two numbers without actually calculating their cubes. a³ + b³ formula is also used to find the factors of a³ + b³. This helps to simplify various algebraic expressions.

In this article, we will learn about, the a³ + b³ formula, factors of a³ + b³, examples, and others in detail.

What is a³ + b³ Formula?

a³ + b³ formula is the basic algebra formula that gives the sum of two cubes without actually calculating the cubes.

Suppose we want to solve 12³ + 8³ without actually finding 12³ and 8³ then we use the a³ + b³ formula to find 12³ + 8³, without calculating the cubes of 8 and 12. The a³ + b³ formula is used to solve and simplify algebraic problems, it is also helpful in solving trigonometric problems, and others. The a³ + b³ formula is given in the image added as,

a³ + b³ Definition

a3 + b3 is a formula that is used to factorize a3 + b3 or it is used to find the sum of cubes of two numbers without actually calculating the cubes. It is derived from the expansion of (a + b)3 and is given as,

(a + b)³ = a³ + b³ + 3ab(a + b)

⇒ a³ + b³ = (a + b)³ – 3ab(a + b)

⇒ a³ + b³ = (a + b){(a + b)² – 3ab}

⇒ a³ + b³ = (a + b){a² + b² + 2ab – 3ab}

⇒ a³ + b³ = (a + b)(a² + b² – ab)

Factor of a³ + b³

To find the factor of a³ + b³ we use, the a³ + b³ formula added above, i.e. a³ + b³ = (a + b)(a² + b² -ab). This formula also gives the factor of a³ + b³. Factors of any number or polynomial are numbers or polynomials that on multiplying give the following number or polynomial. So by observing the a³ + b³ formula we can say that factors of a³ + b³ are,

Factors of a³ + b³ = (a + b)(a² – ab + b²)

What is Sum of Cubes Formula?

Sum of Cubes Formula, also known as the a³ + b³ formula, serves the purpose of calculating the combined value of two specified cubes and assisting in their factorization. The expression for the Sum of Cubes Formula is given below as,

a³ + b³ = (a + b)(a² – ab + b²)

where,

• a is the First Variable
• b is the Second Variable

This formula gives the sum of cube of two numbers without actually calculating the cubes.

a³ + b³ Formula Proof

The sum of cube formula or a³ + b³ formula is proved by solving the RHS part of the formula and then simplifying the same to make it equal to the LHS. The a³ + b³ formula is,

a³ + b³ = (a + b)(a² – ab + b²)

RHS = (a + b)(a² – ab + b²)

Using the Distributive Property of Multiplication, we have

RHS = a(a² – ab + b²)) + b(a² – ab + b²)

⇒ RHS = a³ – a²b + ab² + a²b – ab² + b³

⇒ RHS = a³ – a²b + a²b + ab² – ab² + b³

⇒ RHS = a³ – 0 + 0 + b³

⇒ RHS = a³ + b³

⇒ RHS = LHS

Hence proved.

(a + b)³ Formula

(a + b)³ formula is the formula that is used to expand the cube of sum of two numbers and the formula for the same is,

(a + b)³ = a³ + b³ + 3ab(a + b)

a³ + b³ vs a³ – b³

Both formulas of a³ + b³ and a³ – b³ deal with cubes, but in different ways. The key differences between the a³ + b³ formula and the a³ – b³ formula are listed in the following table:

Read More,

• Algebraic Identites
• Like and Unlike Algebraic Terms
• Difference of Squares

Examples on Sum of Cubes Formula

Example 1: Factorize 343a³ + 216 using sum of cubes.

Solution:

Given,

343a³ + 216 = (7a)³ + (6)³

Since, a³ + b³ = (a + b)(a² – ab + b²)

⇒ 343a³ + 216  = (7a + 6)[(7a)² – (7a)(6) + (6)²]

⇒ 343a³ + 216 = (7a + 6)[49a² – 42a + 36]

Example 2: Factorize 8p³ + 27 using sum of cubes.

Solution:

Given,

8p³ + 27 = (2p)³ + (3)³

Since, a³ + b³ = (a + b)(a² – ab + b²)

⇒ 8p³ + 27 = (2p + 3)[(2p)² – (2p)(3) + (3)²]

⇒ 8p³ + 27 = (2p + 3)[4p² – 6p + 9]

Example 3: Factorize 27t³ + 125 using the sum of cubes.

Solution:

Given,

27t³ + 125 = (3t)³ + (5)³

Since, a³ + b³ = (a + b)(a² – ab + b²)

⇒ 27t³ + 125 = (3t + 5)[(3t)² – (3t)(5) + (5)²]

⇒ 27t³ + 125 = (3t + 5)[9t² – 15t + 25]

Example 4: Factorize 64s³ + 125 using sum of cubes.

Solution:

Given,

64s³ + 125 = (4s)³ + (5)³

Since, a³ + b³ = (a + b)(a² – ab + b²)

⇒ 64s³ + 125  = (4s + 5)[(8s)² – (4s)(5) + (5)²]

⇒ 64s³ + 125 = (8s + 5)[64s² – 20s + 25]

Example 5: Factorize 512 + 729v³ using the sum of cubes formula.

Solution:

512 + 729v³ = (8)³ + (9v)³

Since, a³ + b³ = (a + b)(a² – ab + b²)

⇒ 512 + 729v³  = (8 + 9v)[(8)² – (8)(9v) + (9v)²]

⇒ 512 + 729v³  = (8 + 9v)[64 – 72v + 729v²]

Practice Problems on a³ + b³ Formula

Q1. Find the factor of 8x³ + 27y³

Q2. Factorize a⁶ + 64c³

Q3. Facorize a³ + 1331c³

Q4, Find the factors of y³ + 643z³

a³ + b³ Formula: FAQs

1. What is Expansion Formula of a³ + b³?

The expansion formula of a³ + b³ is given as,

a³ + b³ = (a + b)(a² – ab + b²)

2. What is Difference of Square Formula?

The difference of square formula is the formula in mathematics that is used to find the difference of two squares without actually calculating the squares. The difference of square formula is given as,

a² – b² = (a + b)(a – b)

3. What is Use of a³ + b³ Formula?

The a³ + b³ formula is used to find the sum of two cubes without actually calculating the cubes and it helps to simplify algebraic problems. It is also used to solve trigonometric problems and others.

4. What is Sum of First ‘n’ Cubes?

The sum of first n cubes is given by the formula,

1³ + 2³ + 3³ + … + n³ = n².(n + 1)²/4

5. What is Sum of Cubes from 1 to 100?

The sum of cubes of 1 to 100 is 6,376,625 which can be represented as:

1³ + 2³ + 3³ + . . .  + 100³ = 6,376,625

6. What are Factors of a³ + b³?

The factors of a³ + b³ is given as,

a³ + b³ = (a + b)(a² + b² – ab)

7. What is Formula for a³ + b³?

The formula for a³ + b³ is given as, a³ + b³ = (a + b)(a² – ab + b²)