a3−b3 formula is a fundamental concept in algebra and is often used in competitive exams to test a candidate's understanding of the subject. By having a thorough knowledge of the formula, candidates can solve problems related to algebra more efficiently and accurately.
a3-b3 Formula Examples
Let's consider an example to demonstrate the application of this formula.
Suppose we have a = 5 and b = 2. We can substitute these values into the a3-b3 formula
a3−b3 = (a - b)(a2 + b2 + ab)
(5 - 2)(52 + 5 x 2 + 22)
(3)(25 + 10 + 4)
(3)(39) = 117
Steps to derive the a3-b3 formula
Let’s derive the formula for a3−b3 using the (a−b)3 formula
As we know, (a−b)3 = a3−b3 - 3ab(a - b)
Shift a3 and - b3 to the left-hand side and shift all other terms to the right-hand side to get:
a3−b3 = (a - b)3 + 3ab(a - b)
Taking common terms (a-b) out we get the following
a3−b3 = (a - b)[(a - b)2 + 3ab]
As we know (a - b)2 = (a2 - 2ab + b2)
So, we are replacing (a - b)2 in the above expression with (a2 - 2ab + b2). It can be written as:
a3−b3 = (a - b)[a2 - 2ab + b2 + 3ab]
On simplifying further, we get
a3−b3 = (a - b)(a2 + b2 + ab)
Therefore, the formula of a3−b3 is a3−b3 = (a−b)(a2 + b2 + ab)
What is the Formula of a3−b3?
This algebraic formula is used to calculate the difference between two cubes. It can easily be derived by using the (a −b)3 and the a3−b3 formula is (a−b)(a2 + b2 + ab).
The formula of a3+b3 has been mentioned in the polynomial chapter in mathematics. Using these formulas complex calculations can be solved easily. To add the algebraic expressions, we can use the a3+b3 formula or the a3+b3+c3 formula. Similarly, for the subtraction of the algebraic expressions, the a3-b3 formula can be used.
The identities related to the a3+b3 formula have been shown in this post with examples or solutions to the questions based on these formulas.
The formula for a3 b3 for the different mathematical operations such as addition and subtraction are as follows:
- The (a-b)3 formula = a3 - b3 - 3ab (a-b).
- The formula of a3-b3 = (a2 + ab + b2)(a - b).
- The (a + b)3 formula = a3 + b3 + 3ab(a + b)
- The formula for a3+b3 = (a2 - ab + b2)(a + b).
- The a3 + b3 + c3 formula = (a + b + c) (a2 + b2 + c2 - ab - bc - ca) + 3abc
a3+b3 Formula Examples
Examples based on the a3b3 formula for addition and subtraction are given below:
Let a = 1, b =2, and c = 3. Now put these values in the above-mentioned formulas to get the answer.
- The answer for the a3+b3 formula = (12 - 1x2 + 22)(1+2) = (1-2+4)(3) = 15.
- The solution for (a + b)3 formula = 13 + 23 + 3 x 1 x 2(1+2) = 1+8+18 = 27.
- The solution for a3 + b3 + c3 formula = (1+2+3)(12 + 22 + 32 - 1x2 - 2x3 - 3x1) = 18.
Steps to derive the a3+b3 formula
The step-by-step process to derive the formula has been explained below:
We know that (a + b)3 =(a + b) (a + b)2.
Then, (a + b)3 = (a + b)(a2 + 2ab + b2)
=a(a2 + 2ab + b2) + b(a2 + 2ab + b2)
= a3 + 2a2 b + ab2 + a2b + 2ab2 + b3
= a3 + 3a2 b + 3ab2 + b3
= a3 + b3 + 3ab(a + b)
From the above expression, (a + b)3 = a3 + b3 + 3ab(a + b). Now the formula of a3+b3 can be determined in the following ways:
- (a + b)3 = a3 + b3 + 3ab(a + b)
- a3 + b3 = 3ab(a + b) - (a + b)3
Thus, the formula of a3 + b3 = 3ab(a + b) - (a + b)3 or (a2 - ab + b2)(a + b).
What is the Formula of a3+b3?
This algebraic formula is used to calculate the sum of two cubes. You can easily derive it by using (a + b)3 and its formula is:
a3 + b3 = 3ab(a + b) - (a + b)3 or a3 + b3 = (a2 - ab + b2)(a + b)
What is the Formula of a3+b3 and a3−b3?
The formula of a3+b3 and a3-b3 is very important to solve algebra questions in Mathematics. a3-b3 algebraic formula is used for the calculation of the difference between two cubes and its a3−b3 formula is = (a−b)(a2 + b2 + ab). On the other hand, the formula of a3+b3 is a3 + b3 = 3ab(a + b) - (a + b)3 or a3 + b3 = (a2 - ab + b2)(a + b).
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