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(a-b)^3 Formula
The (a - b)^3 formula is used to find the cube of a binomial. This formula is also used to factorize some special types of trinomials. This formula is one of the algebraic identities. The (a-b)^3 formula is the formula for the cube of the difference of two terms. This formula is used to calculate the cube of the difference of two terms very easily and quickly without doing complicated calculations. Let us learn more about (a-b)^3 formula along with solved examples.
What Is the (a - b)^3 Formula?
The (a-b)^3 formula is used to calculate the cube of a binomial. The formula is also known as the cube of the difference between two terms. 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
= a3 - 3ab(a-b) - b3
Therefore, (a - b)3 formula is:
(a - b)3 = a3 - 3a2b + 3ab2 - b3
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 = 95
Answer: x3 - y3 = 95
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 - b)3 Formula?
(a - b)3 formula is read as a minus b whole cube. Its expansion is expressed as (a - b)3 = a3 - 3a2b + 3ab2 - b3
What Is 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 - b3How 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 and 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.
How To Use the (a - b)3 Formula?
The following steps are followed while using (a - b)3 formula.
- Firstly observe the pattern of the numbers whether the numbers have whole ^3 as power or not.
- Write down the formula of (a - b)3
- (a - b)3 = a3 - 3a2b + 3ab2 - b3
- Substitute the values of a and b in the (a - b)3 formula and simplify.
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