The a-b whole cube formula, i.e. (a-b)3 formula, is used to lớn find the cube of the difference between two terms. This formula is also used to lớn factorise some types of trinomials. The a-b whole cube formula is one of the important algebraic identities. Generally, the (a-b)3 formula is used to solve the problems quickly without undergoing any complicated calculations. In this article, we are going khổng lồ learn the a-b whole cube formula, derivation và examples in detail.
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A-B Whole Cube Formula
(a-b)3 formula is used khổng lồ calculate the cube of a binomial. (a-b)3 is nothing but (a-b)(a-b)(a-b).
The a-b whole cube formula is given by:
(a – b)3 = a3 – 3a2b + 3ab2 – b3 |
(a-b)^3 Formula Derivation
To derive the formula for (a-b)3, we have to lớn multiply (a-b) thrice by itself. (i.e) (a-b)(a-b)(a-b). Go through the below steps to lớn find the formula for (a-b)3.
Derivation:
(a-b)3 = (a-b)(a-b)(a-b)
(a-b)3 = (a2-2ab+b2) (a-b) (a-b)3 = a3-2a2b+ab2-a2b+2ab2-b3 (a-b)3 = a3-3a2b+3ab2– b3 Therefore, the formula for (a-b)3 is a3-3a2b+3ab2– b3. The above formula can also be written as: (a-b)3 = a3-3ab(a-b) – b3. Example 1: Solve the expression (x-2y)3. Solution: Given expression: (x-2y)3. We know that (a-b)3 = a3-3a2b+3ab2– b3 In the expression (x-2y)3, a = x & b = 2y. Now, substitute the value in the a-b whole cube formula, we get (x-2y)3 = x3– 3(x)2(2y) + 3(x)(2y)2 – (2y)3 (x-2y)3 = x3 – 6x2y+12xy2 – 8y3. Hence, (x-2y)3 = x3 – 6x2y+6xy2 – 8y3. Example 2: Solve the expression: (2x – 7y)3 Solution: Given: (2x – 7y)3. As we know, (a-b)3 = a3-3a2b+3ab2– b3 Here, a = 2x & b = 7y By substituting the values in the algebraic identity, we get (2x – 7y)3 = (2x)3 – 3(2x)2(7y) + 3(2x)(7y)2 – (7y)3 (2x -7y)3 = 8x3 – 84x2y +294xy2 – 343y3 Therefore, (2x – 7y)3 = 8x3 – 84x2y +294xy2 – 343y3. To learn more Maths-related concepts và formulas, stay tuned with BYJU’S – The Learning App and explore more videos. The (a - b)^3 formula is used lớn find the cube of a binomial which is made up of the difference of two terms. It says (a - b)3 = a3 - 3a2b + 3ab2 - b3. This is one of the algebraic identities. This formula is used to calculate the cube of the difference between two terms very easily và quickly without doing complicated calculations. Let us learn more about a minus b whole cube formula along with solved examples.Also read: Examples on (a-b)^3
Video Lesson on Algebraic Expansion
What is the (a - b)^3 Formula?
The (a-b)^3 formula is used lớn calculate the cube of a binomial. The formula is also known as the cube of the difference between two terms. According to lớn "a minus b whole cube formula",
(a - b)3 = a3 - 3a2b + 3ab2 - b3 (or) a3 - b3 - 3ab (a - b)
We can derive this formula in two ways:
Method 1: By expanding (a - b)3 as (a - b) (a - b) (a - b).Method 2: By using the formula of (a + b)3Derivation 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 lớn 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.
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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 và 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
FAQs on (a - b)^3 Formula
What is the Expansion of a Minus b Whole Cube Formula?
a minus b whole cube formula is denoted by (a - b)3. Its expansion is expressed as (a - b)3 = a3 - 3a2b + 3ab2 - b3.
What s the (a - b)3 Formula in Algebra?
The (a - b)3 formula is also known as one of the important algebraic identities. It is read as a minus b whole cube. Its (a - b)3 formula is expressed as (a - b)3 = a3 - 3a2b + 3ab2 - b3.
What is the Formula of a Minus b Minus c Whole Cube?
The formula for (a - b - c)3 is (a - b - c)3 = a3 - b3 - c3 + 3 (a - b) (- b - c) (-c + a).
How to Simplify Numbers Using the (a - b)3 Formula?
Let us understand the use of the (a - b)3 formula with the help of the following example.
Example: Find the value of (20 - 5)3 using the (a - b)3 formula.
To find: (20 - 5)3
Let us assume that a = 20 và b = 5.
We will substitute these in the formula of (a - b)3.
(a - b)3 = a3 - 3a2b + 3ab2 - b3
(20-5)3 = 203 - 3(20)2(5) + 3(20)(5)2 - 53
= 8000 - 6000 + 1500 - 125
= 3375
Answer: (20 - 5)3 = 3375.
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How lớn Use the (a - b)3 Formula?
To use a minus b whole cube formula, first write the formula (a - b)3 = a3 - 3a2b + 3ab2 - b3. Then substitute the values of a & b here. Finally, simplify.