What is a3+b3
The a3 +b3 formula is an algebraic identity which is read as a cube plus b cube. The a3 +b3 formula is used to find factorise cubes of binomials.
a3+b3 Formula
Basic:
The a3 +b3 formula is an algebraic identity which is read as a cube plus b cube. The a3 +b3 formula is used to find factorise cubes of binomials. Thea3 +b3 formula helps us find the sum of cubes of two numbers easily without calculating the cubes of numbers. The a3 +b3 formula is applicable for all values of a and b. Sometime it is also called as ‘the sum of cubes’ formula.
We will be discussing aspects on the a3 +b3 formula, along with solved examples, and understand the identity involved.
What Is a3 +b3 Formula?
The a3 +b3 = (a+b)(a2-ab+b2) which can verified as well as can be derived from the formula (a+b)3. Lets discuss both aspects.
To verify a3+b3 = (a+b)(a2-ab+b2)
Solving the RHS
(a +b) (a2 - ab + b2)
Using binomial multiply with trinomial
= 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
LHS we have a3+b3
Therefore, LHS=RHS.
To prove by using (a+b)3 identity
Solved Example for a3 +b3 formula
Find the value of 103 + 23 by using the a3+b3 formula.
Solution: To find: 103 + 23.
Let us assume that a = 10 and b = 2
We will substitute these in the formula of a3 + b3 i.e.,
a3 + b3 = (a + b) (a2 - ab + b2)
103+23 = (100+8) = 108
Get the List of Maths formulas.
The (a3+b3) Formula is an important algebraic identity used to factorize the sum of two cubes. It is widely used in algebra, polynomial simplification, and equation solving. Students learn this identity to simplify expressions quickly and solve higher mathematics problems efficiently.
The formula is:
a3+b3=(a+b)(a2-ab+b2)
This identity helps express the sum of two cubes as the product of two algebraic expressions.
Meaning of the Formula
In the formula:
- (a3) means cube of the first term
- (b3) means cube of the second term
- (a+b) is the common factor
- (a2-ab+b2) is the quadratic factor
The formula is mainly used for factorisation rather than expansion.
How to Use (a3+b3) Formula
Follow these steps:
- Identify both cube terms.
- Find their cube roots.
- Write the sum of cube roots in one bracket.
- Write the quadratic expression in the second bracket.
- Simplify if needed.
Applications of (a3+b3) Formula
The formula is used in:
- Algebraic factorization
- Polynomial simplification
- Solving equations
- Competitive mathematics
- Higher algebra
- Coordinate geometry
It helps reduce lengthy calculations and makes algebraic problems easier.
Important Points to Remember
- The formula is used for the sum of cubes only.
- The middle sign in the second bracket is always negative.
- First bracket contains the sum of cube roots.
- The second bracket contains square, product, and square terms.
- Practice helps avoid sign mistakes.
Difference Between (a3+b3) and (a3-b3)
| Formula Type | Identity |
|---|---|
| Sum of Cubes | a3+b3=(a+b)(a2-ab+b2) |
| Difference of Cubes | a3-b3=(a-b)(a2+ab+b2 |
Conclusion
The (a3+b3) Formula is a very useful algebraic identity for factorizing cubic expressions. It simplifies mathematical calculations and helps students solve algebra problems quickly. By understanding the formula structure and practicing examples regularly, students can improve their problem-solving skills and strengthen their algebra concepts.
FAQs on a³ + b³ Formula
The formula for a³ + b³ is called the sum of cubes identity. It is written as:
a3+b3=(a+b)(a2-ab+b2)
This algebraic identity helps in factorization and simplification of cube expressions. It is widely used in mathematics to solve equations, simplify algebraic problems, and understand polynomial identities. For example, if a = 2 and b = 3, then:
2³ + 3³ = 8 + 27 = 35
Using the formula:
(2 + 3)(2² − 2×3 + 3²) = 5(4 − 6 + 9) = 5 × 7 = 35
Both methods give the same result, proving the formula is correct.
The a³ + b³ formula is used in algebra, polynomial factorization, competitive exams, and higher mathematics. Students use it to simplify expressions and solve equations involving cubes. It is also useful in factorizing large algebraic terms quickly without lengthy calculations.
For example, expressions like x³ + 8 or 27a³ + b³ can be simplified using the sum of cubes formula. In schools, this identity is commonly taught to improve problem-solving speed and understanding of algebraic concepts. It is especially important in board exams and entrance test preparation because many questions are based on algebraic identities and factorization methods.