A Cube Minus B Cube Formula – Examples | a^3

The a– b3 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 be discussing the various aspects of the a– b3 formula, along with solved examples, and understand the identity involved.

What Is a^3-b^3 Formula?

The a– b3 formula can be verified, by multiplying (a – b) (a2 + ab + b2) and see whether you get a3 – b3. The a– b3 formula or the difference of cubes formula is explained below:

a– b3 Formula = a– b= (a – b) (a2 + ab + b2)

You can remember these signs using the following trick.

a cube minus b cube formula

Let us learn the a– b3 formula with a few solved examples.

Proof of a3-b3 Formula

Let us verify the a cube minus b cube formula. To prove that a– b= (a – b) (a2 + ab + b2) we need to prove here LHS = RHS. Lets begin with the following steps.

LHS = a– b3

On Solving RHS side we get,

= (a – b) (a2 + ab + b2)

On multiplying the a and b separately with (a2 + ab + b2) we get

= a (a2 + ab + b2) – b(a2 + ab + b2)

= a3 + a2b + ab2 – a2b – ab– b3

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= a3 + a2b – a2b + ab2– ab– b3

= a3 – 0 – 0 – b3

= a3 – b3

Hence proved, LHS = RHS

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Examples on a^3 – b^3 Formula

Example 1: Find the value of 1083 – 83 using the a^3 – b^3 formula.

Solution:

To find: 1083 – 83.

Let us assume that a = 108 and b = 8.

We will substitute these in the formula of a3 – b3.

a– b= (a – b) (a2 + ab + b2)

1083-83 = (108-8)(1082 + (108)(8) + 82)

= (100) (11664+864+64)

= (100)(12592)

=1259200

Answer: 1083 – 83 = 1,259,200.

Example 2: Factorize the expression 27x3 – 125 using a^3-b^3 formula.

Solution: 

To factorize: 27x3 – 125.

We will use the a3 – bformula to factorize this.

We can write the given expression as

27x3 – 125 = (3x)3 – 53

We will substitute a = 3x and b = 5 in the formula of a3 – b3.

a– b= (a – b) (a2 + ab + b2)

(3x)3-53 =(3x-5)((3x)2+(3x)(5)+52)

= (3x-5) (9x2+15x+25)

Answer: 27x3 – 125 = (3x – 5) (9x2 + 15x + 25).

Example 3: Simplify 193 – 203 using a cube minus b cube formula.

Solution: To find 193 – 203

Let us assume  a = 19 and b = 20

Using formula a– b= (a – b) (a2 + ab + b2)

We will substitute these in the a3 – b3 formula

a– b= (a – b) (a2 + ab + b2)

193-203 = (19-20)(192 + (19)(20)+202)

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

= (-1)(1141)

= -1141

Answer: 193 – 203 = -1141.

FAQ’s on a^3 – b^3 Formula

What Is the Expansion of a– b3 Formula?

a– b3 formula is read as a cube minus b cube. Its expansion is expressed as a– b3 = (a – b) (a2 + ab + b2).

See also  Mixed Fractions - Definition, Conversions, Examples.

What Is the a– b3 Formula in Algebra?

The a– b3 formula is also known as one of the important algebraic identiy. It is read as a cube minus b cube. Its a– b3 formula is expressed as a– b3 = (a – b) (a2 + ab + b2).

How To Simplify Numbers Using the a– b3 Formula?

Let us understand the use of the a– b3 formula with the help of the following example.

Example: Find the value of 103 – 23 using the a– b3 formula.

To find: 103 – 23

Let us assume that a = 10 and b = 2.

We will substitute these in the formula of a3 – b3.

a– b= (a – b) (a2 + ab + b2)

103-23 = (10-2)(102 + (10)(2)+22)

= (8) (100+20+4)

= (8)(124)

=992

Answer: 103 – 23 = 992.

How To Use the a– b3 Formula Give Steps?

The following steps are followed while using a– b3 formula.

    • Firstly observe the pattern of the numbers whether the numbers have ^3 as power or not.
    • Write down the formula of a3 – b3
    • a– b= (a – b) (a2 + ab + b2)
    • substitute the values of a and b in the a– b3 formula and simplify.

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