(A - B)^3 Formula | A Minus B Whole Cube Formula with Examples

The (a − b)³ formula is a very helpful rule in math. It helps you find the answer when you multiply a subtraction problem by itself three times. Instead of doing long multiplication, you can just use this short rule. In this article, we will learn the a b 3 formula expansion, see how it works, and look at easy examples.

What is the (a - b)³ Formula?

In simple words, the (a - b)³ formula is the "A Minus B Whole Cube" rule. We use it when we see two numbers being subtracted and then raised to the power of 3.

The Formula

There are two ways to write the (a − b)³ formula:

1. (a – b)³ = a³ – 3a²b + 3ab² – b³

2. (a – b)³ = a³ – b³ – 3ab(a – b)

Both are the same! The second one is just a shorter way to write it.

Read More: (a + b)³ Formula

Steps to Calculate (a - b)^3 Formula

Multiplying the Brackets

To find (a - b)³, we multiply (a - b) by itself three times:

(a - b)³ = (a - b) × (a - b) × (a - b)

1. First, multiply the first two: (a - b) × (a - b) = a² - 2ab + b².

2. Now, multiply that answer by the last (a - b).

3. After you multiply everything and group the same parts together, you get:
   = a³ – 3a²b + 3ab² – b³

This is the a b 3 formula proof!

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Understanding the Pattern

It is easy to remember the formula if you look at the pattern:

• The Numbers: The numbers in front of the parts are always 1, 3, 3, and 1.

• The Powers: The power of 'a' goes down (3, 2, 1, 0) and the power of 'b' goes up (0, 1, 2, 3).

• The Signs: The signs go back and forth (+ , - , + , -).

Comparing Plus and Minus Formulas

Read More: A^2-B^2 Formula

(a - b)³ Formula Examples

Let’s look at some examples.

Example 1: Solve (x - 3)³

Solution:

Here, a = x and b = 3.

Using the formula: (a – b)³ = a³ – 3a²b + 3ab² – b³

• x³ - 3(x²)(3) + 3(x)(3²) - 3³

• x³ - 9x² + 3(x)(9) - 27

• Final Answer: x³ – 9x² + 27x – 27

Example 2: Solve (10 - 2)³

Solution:

Here, a = 10 and b = 2.

• (10)³ - 3(10²)(2) + 3(10)(2²) - (2)³

• 1000 - 3(100)(2) + 3(10)(4) - 8

• 1000 - 600 + 120 - 8

• Final Answer: 512
  (Check: 10 - 2 = 8, and 8 × 8 × 8 = 512. It works!)

 Derivation of (a - b)^3: Method 1

To find the formula of (a - b)3, we will just multiply (a - b) (a - b) (a - b).

(a - b)3 = (a - b)(a - b)(a - b)

= (a2 - 2ab + b2)(a - b)

= a3 - a2b - 2a2b + 2ab2 + ab2 - b3

= a3 - 3a2b + 3ab2 - b3 (or)

= a3 - b3 - 3ab (a - b)

Therefore, (a - b)3 formula is:

(a - b)3 = a3 - 3a2b + 3ab2 - b3

Hence proved.

Derivation of (a - b)^3: Method 2

We use the formula of (a + b)3 to derive the formula of a minus b whole cube. We know that

(a + b)3 = a3 + 3a2b + 3ab2 + b3

Replace b with -b on both sides of this formula:

(a + (-b))3 = a3 + 3a2(-b) + 3a(-b)2 + (-b)3

This results in (a - b)3 = a3 - 3a2b + 3ab2 - b3.

Hence derived.

Read More: Brackets in Maths

Examples on (a - b)^3 Formula

Example 1: Solve the following expression using (a - b)3 formula:
(2x - 3y)3

Solution: To find: (2x - 3y)3

Using (a - b)3 Formula,

(a - b)3 = a3 - 3a2b + 3ab2 - b3

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

= 8x3 - 36x2y + 54xy2 - 27y3

Answer: (2x - 3y)3 = 8x3 - 36x2y + 54xy2 - 27y3

Example 2: Find the value of x3 - y3 if x - y = 5 and xy = 2 using (a - b)3 formula.

Solution: To find: x3 - y3

Given:

x - y = 5

xy = 2

Using (a - b)3 Formula,

(a - b)3 = a3 - 3a2b + 3ab2 - b3

Here, a = x; b = y

Therefore,

(x - y)3 = x3 - 3 × x2 × y + 3 × x × y2 - y3

(x - y)3 = x3 - 3x2y + 3xy2 - y3

53 = x3 - 3xy(x - y) - y3

125 = x3 - 3 × 2 × 5 - y3

x3 - y3 = 155

Answer: x3 - y3 = 155

Example 3: Solve the following expression using (a - b)3 formula: (5x - 2y)3

Solution: To find: (5x - 2y)3

Using (a - b)3 Formula,

(a - b)3 = a3 - 3a2b + 3ab2 - b3

= (5x)3 - 3 × (5x)2 × 2y + 3 × (5x) × (2y)2 - (2y)3

= 125x3 - 150x2y + 60xy2 - 8y3

Answer: (5x - 2y)3 = 125x3 - 150x2y + 60xy2 - 8y3

Common Mistakes to Watch Out For

• Wrong Signs: Do not make all the signs plus. It must go plus, minus, plus, minus.

• Missing the 3: Do not forget the number '3' in the middle parts.

• Multiplying wrong: Remember that (2x)³ means 2 × 2 × 2 and x × x × x, which is 8x³.

Tips for Success

To be great at using this formula, follow these simple tips:

• Sign Check: Always check your plus and minus signs. The minus signs should always be with the second and last parts.

• Use Brackets: When putting numbers into the formula, use brackets. This stops you from making mistakes when squaring or cubing numbers like 2x.

• Know Your Cubes: Learn small cubes like 1³=1, 2³=8, 3³=27, and 4³=64. This helps you find the answer much faster.

• Keep Practicing: Solve one new problem every day to keep your brain sharp! 

Also Read: Construction in Maths

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