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# a^3 - b^3 Formula

The a^{3} - b^{3} formula is called the difference of cubes (of two numbers) formula. The a cube minus b cube formula is used to find the difference between the two cubes without actually calculating the cubes. Also, it is used to factorize the binomials of cubes. In this section, we will discuss the various aspects of the a^3 - b^3 formula, along with solved examples, and understand the identity involved.

## What is a^3 - b^3 Formula?

The a^{3 }- b^{3} formula or the difference of cubes formula is as follows:

**a ^{3 }- b^{3 }= (a - b) (a^{2} + ab + b^{2})**

You can remember these signs using the following trick.

"a cube minus b cube formula" can be verified, by multiplying (a - b) and (a^{2} + ab + b^{2}) and see whether you get a^{3} - b^{3}.

## Proof of a Cube Minus b Cube Formula

Let us verify the a cube minus b cube formula. To prove that a^{3 }- b^{3 }= (a - b) (a^{2} + ab + b^{2}) we need to prove here LHS = RHS. Let's begin with the following steps.

LHS = a^{3 }- b^{3}

On Solving RHS side we get,

= (a - b) (a^{2} + ab + b^{2})

On multiplying a and b separately with (a^{2} + ab + b^{2}) we get

= a (a^{2} + ab + b^{2}) - b(a^{2} + ab + b^{2})

= a^{3} + a^{2}b + ab^{2} - a^{2}b - ab^{2 }- b^{3}

= a^{3} + a^{2}b - a^{2}b + ab^{2}- ab^{2 }- b^{3}

= a^{3} - 0 - 0 - b^{3}

= a^{3} - b^{3}

Hence proved, LHS = RHS

**☛ Also Check:** a^3+b^3 Formula

## Examples on a^3 - b^3 Formula

Let us learn the a^{3} - b^{3} formula with a few solved examples.

**Example 1:** Find the value of 108^{3} - 8^{3} using a^3^{ }- b^3 formula.

**Solution:**

To find: 108^{3} - 8^{3}.

Let us assume that a = 108 and b = 8.

We will substitute these in the formula of a^{3} - b^{3}.

a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})

108^{3} - 8^{3} = (108 - 8) (108^{2} + (108)(8) + 8^{2})

= (100) (11664+864+64)

= (100) (12592)

=1259200

**Answer:** 108^{3} - 8^{3} = 1,259,200.

**Example 2:** Factorize the expression 27x^{3} - 125 using a^3 - b^3 formula.

**Solution:**

To factorize: 27x^{3} - 125.

We will use the a^{3} - b^{3 }formula to factorize this.

We can write the given expression as

27x^{3} - 125 = (3x)^{3} - 5^{3}

We will substitute a = 3x and b = 5 in the formula of a^{3} - b^{3}.

a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})

(3x)^{3} - 5^{3} = (3x - 5) ((3x)^{2} + (3x)(5) + 5^{2})

= (3x - 5) (9x^{2 }+ 15x + 25)

**Answer:** 27x^{3} - 125 = (3x - 5) (9x^{2} + 15x + 25).

**Example 3:** Simplify 19^{3} - 20^{3} using a cube minus b cube formula.

**Solution:** To find 19^{3} - 20^{3}

Let us assume a = 19 and b = 20

Using formula a^{3 }- b^{3 }= (a - b) (a^{2} + ab + b^{2})

We will substitute these in the a^{3} - b^{3} formula

a^{3 }- b^{3 }= (a - b) (a^{2} + ab + b^{2})

19^{3 }- 20^{3} = (19 - 20) (19^{2} + (19)(20) + 20^{2})

= (-1) (361 + 380 + 400)

= (-1) (1141)

= -1141

**Answer:** 19^{3} - 20^{3} = -1141.

## FAQ's on a^3 - b^3 Formula

### What is the Expansion of a^{3 }- b^{3} Formula?

**a ^{3 }- b^{3} formula** is read as a cube minus b cube. Its expansion is expressed as a

^{3 }- b

^{3}= (a - b) (a

^{2}+ ab + b

^{2}).

### What is the a^{3 }- b^{3} Formula in Algebra?

The a^{3 }- b^{3} formula is also known as one of the important algebraic identiies. It is read as a cube minus b cube. a^{3 }- b^{3} formula is a^{3 }- b^{3} = (a - b) (a^{2} + ab + b^{2}).

### How to Simplify Numbers Using a cube^{ }- b cube Formula?

Let us understand the use of the a^{3 }- b^{3} formula with the help of the following example.

**Example:** Find the value of 10^{3} - 2^{3} using the a^{3 }- b^{3} formula.

To find 10^{3} - 2^{3}, let us assume that a = 10 and b = 2.

We will substitute these in the formula of a^{3} - b^{3}.

a^{3 }- b^{3 }= (a - b) (a^{2} + ab + b^{2})

10^{3 }- 2^{3} = (10 - 2) (10^{2} + (10)(2) + 2^{2})

= (8) (100 + 20 + 4)

= (8)(124)

= 992

**Answer:** 10^{3} - 2^{3} = 992.

### What are the Applications of a^3 - b^3 Formula?

The a cubed minus b cubed formula is used to:

- Factorize algebraic expressions. Example: x
^{3}- 27 = x^{3}- 3^{3}= (x - 3) (x^{2}+ 3x + 9) - Simplify trigonometric expressions. Example: sin
^{3}x - cos^{3}x = (sin x - cos x) (sin^{2}x + sin x cos x + cos^{2}x) = (sin x - cos x) (1 + sin x cos x)

### What are a^3 - b^3 and a^3 + b^3 formulas?

- a^3 - b^3 = (a - b) (a^2 + ab + b^2)
- a^3 + b^3 = (a + b) (a^2 - ab + b^2)

Note that in each of these formulas, the sign between a and b is the same sign as the sign on the left side. ab carries the opposite sign and b^2 is always positive.

### How to Use the a^{3 }- b^{3} Formula Give Steps?

The following steps are followed while using a cube minus b cube formula.

- To begin with, observe the pattern of the numbers whether the numbers have ^3 as power or not.
- Further, Write down the formula of a^3 - b^3
^{ }: a^{3 }- b^{3 }= (a - b) (a^{2}+ ab + b^{2}) - Finally, substitute the values of a and b in the a cubed
^{ }- b cubed formula and simplify.

### What is a Cube Minus b Cube Minus c Cube Formula?

We derive the formula for a cube minus b cube minus c cube by using the formula: a^{3} + b^{3} + c^{3} - 3abc = (a + b + c)(a^{2} + b^{2} + c^{2} - ab - bc - ca) by substituting b = -b and c = -c into it. Then we get

a^{3} + (-b)^{3} + (-c)^{3} - 3a(-b)(-c) = (a - b - c)(a^{2} + (-b)^{2} + (-c)^{2} - a(-b) - (-b)(-c) - (-c)a)

a^{3} - b^{3} - c^{3} - 3abc = (a - b - c)(a^{2} + b^{2} + c^{2} + ab - bc + ca)

Adding 3abc on both sides:

a^{3} - b^{3} - c^{3} = (a - b - c)(a^{2} + b^{2} + c^{2} + ab - bc + ca) + 3abc

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