A Cube Minus B Cube Formula with Examples

A Cube Minus B Cube Formula - The a3-b3 formula (a cube minus b cube) denotes how the cube value of one number differs from that of another. The formula of a3-b3 (a cube minus b cube) helps to simplify complex algebraic expressions, making problem solving easier. 

This a^3-b^3 formula (A Cube Minus B Cube formula) finds application in practical-based problems, architecture, and engineering, where quick algebraic calculations are required for correct measurements. 

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What is A Cube Minus B Cube?

A cube minus B cube - a3-b3 formula is a mathematical expression representing the difference between the cubes of two numbers, denoted by A and B, respectively.

If a and b are two numbers, then the difference of their cubes is expressed by the mathematical expression a3 - b3

The formula of a cube minus b cube is the expansion or simplified form of this expression, which helps calculate the difference of cubes of these numbers conveniently and quickly. The best part is that you don’t have to calculate the cube of the numbers to find the difference.

Formula for a3 - b3

The a^3 - b^3 formula is as follows:

a3 - b3 = (a - b) (a2 + ab + b2)

There is a trick to remember this a cube minus b cube formula as denoted in the image given below.

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Proof of the a3 - b3 Formula

To prove the a3 - b3 formula, we can solve the expression on the RHS side of the above expression and check whether it is equal to a3 - b3.

a3 - b3 = (a - b) (a2 + ab + b2)

here, LHS = a3 - b3 and RHS = (a - b) (a2 + ab + b2)

We have to prove that LHS = RHS

Let’s solve the RHS.

(a - b) (a2 + ab + b2)

= a (a2 + ab + b2) – b (a2 + ab + b2)
= a3 + a2b + ab2 - a2b - ab2 - b3
= a3 + a2b - a2b + ab2- ab2 - b3
= a3 - 0 - 0 - b3
= a3 - b3

So, it is proved that LHS = RHS

Thus, a³ - b³ formula is verified.

Read More: Even Numbers

Use of A Cube Minus B Cube Formula with Examples

The formula of a3 - b3 is used to find the difference of cubes of two numbers, and there is no need to calculate the cube of the respective numbers.

a^3 - b^3 formula example  - Find t example, to find the value of 243 - 183, we can use the formula of a3 - b3.

243 - 183 =

= (24 - 18) (242 + (24) x (18) + 182)

= (6) x (576 + 432 + 324)

= (6) x (1332)

= 7992

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Factorization Using a3 - b3 Formula

Often, we need to factorize algebraic expressions involving the difference in the cube of numbers. In such cases, the formula of a3 - b3 becomes very useful.

For example, we have to factorize the following expression.

125x3 – 27

We will write this expression in the form of a3 - b3.

So, 125x3 – 27

= (5x)3 - (3)3    

Now, using the formula of a3 - b3, we can write:

 Therefore, 125x3 – 27 = (5x -3) (25x2 + 15x + 9)

Read More: Odd Numbers

Steps for Applying a3 - b3 Formula

We must follow the steps below to apply the a3 - b3 formula for an accurate solution to a given problem.

• Look at the algebraic expression to find whether the two numbers can be denoted as a cube of particular numbers.

• Write the expression in a3 - b3   format.

• Identify a and b from the format and replace them in the expression (a - b) (a2 + ab + b2)

• Solve the expression (a - b) (a2 + ab + b2)

What is A Cube Minus B Whole Cube? 

It is an algebraic expression that denotes the cube of the difference of two numbers. If two numbers are a and b, this expression is written as (a – b)3.

There is a formula for (a – b)3 which is expressed as follows:

(a – b)³ = a³ - 3a²b + 3ab² - b³

A Cube minus B Cube Formula Solved Examples

1. Find the value of 623 -553   

Solution:

623 -553  

= (62 - 55) (622 + (62) x (55) + 552)

= (7) x (3844 + 3410 + 3025)

= (7) x (10279)

= 71953

2. Factorize the expression 216p3 -64q3   

Solution:

We know that 216 = 63   and 64 = 43   

So, we can write,

216p3 -64q3   

= (6p)3 – (4q)3     

Here a = 6p and b = 4q

Using the formula a3 - b3 = (a - b) (a2 + ab + b2), we can write:

(6p)3 – (4q)3     

= (6p – 4q) [(6p)2 + (6p) (4q) + (4q)2]

= (6p -4q) (36p2 + 24pq + 16q2)

Therefore, 216p3 -643   = (6p -4q) (36p2 + 24pq + 16q2)

3. Simplify the expression (3x – 2y)3 using a minus b whole cube formula.

Solution:

The a minus b whole cube formula states that (a – b)³ = a³ - 3a²b + 3ab² - b³

Here, a = 3x and b = 2y

So, (3x – 2y)3

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

= 27x³ - 6 (9x²)y + 9x(4y²) - 8y³

= 27x³ - 54x²y + 36xy² - 8y³

4. There are two cube shaped boxes of length 15 cm and 10 cm respectively. Find the difference of their volumes. 

Solution:

We know that the volume of a cube-shaped box is a x a x a = a³, where a is the dimension of each side of the box.

In this case, the volume of first box is (15)³ and the volume of second box is (10)³

So, the difference is the volume of two boxes can be calculated as follows:

(15) ³ - (10) ³

= (15-10) (152 + 15 x 10 + 102 )

= (5) (225 + 150 + 100)

= 5 x 475

= 2375

Therefore, the difference in the volume of two boxes is 2375 cubic cm.

This article simplifies the a³–b³ formula to build stronger problem-solving skills.

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