a3 + b3 Formula
a3 + b3 formula which is also called the sum of cubes formula, in algebra is the fundamental algebra formula that is used to find the sum of cubes of any two numbers without actually calculating their cubes. a3 + b3 formula is also used to find the factors of a3 + b3. This helps to simplify various algebraic expressions.
In this article, we will learn about, the a3 + b3 formula, factors of a3 + b3, examples, and others in detail.
Table of Content
- What is a3 + b3 Formula?
- What is Sum of Cubes Formula?
- a3 + b3 Formula Proof
- Examples on Sum of Cubes Formula
- a3 + b3 Formula β FAQs
What is a3 + b3 Formula?
a3 + b3 formula is the basic algebra formula that gives the sum of two cubes without actually calculating the cubes.
Suppose we want to solve 123 + 83 without actually finding 123 and 83 then we use the a3 + b3 formula to find 123 + 83, without calculating the cubes of 8 and 12. The a3 + b3 formula is used to solve and simplify algebraic problems, it is also helpful in solving trigonometric problems, and others. The a3 + b3 formula is given in the image added as,
a3 + b3 Definition
a3 + b3 is a formula that is used to factorize a3 + b3 or it is used to find the sum of cubes of two numbers without actually calculating the cubes. It is derived from the expansion of (a + b)3 and is given as,
(a + b)3 = a3 + b3 + 3ab(a + b)
β a3 + b3 = (a + b)3 β 3ab(a + b)
β a3 + b3 = (a + b){(a + b)2 β 3ab}
β a3 + b3 = (a + b){a2 + b2 + 2ab β 3ab}
β a3 + b3 = (a + b)(a2 + b2 β ab)
Factor of a3 + b3
To find the factor of a3 + b3 we use, the a3 + b3 formula added above, i.e. a3 + b3 = (a + b)(a2 + b2 -ab). This formula also gives the factor of a3 + b3. Factors of any number or polynomial are numbers or polynomials that on multiplying give the following number or polynomial. So by observing the a3 + b3 formula we can say that factors of a3 + b3 are,
Factors of a3 + b3 = (a + b)(a2 β ab + b2)
What is Sum of Cubes Formula?
Sum of Cubes Formula, also known as the a3 + b3 formula, serves the purpose of calculating the combined value of two specified cubes and assisting in their factorization. The expression for the Sum of Cubes Formula is given below as,
a3 + b3 = (a + b)(a2 β ab + b2)
where,
- a is the First Variable
- b is the Second Variable
This formula gives the sum of cube of two numbers without actually calculating the cubes.
a3 + b3 Formula Proof
The sum of cube formula or a3 + b3 formula is proved by solving the RHS part of the formula and then simplifying the same to make it equal to the LHS. The a3 + b3 formula is,
a3 + b3 = (a + b)(a2 β ab + b2)
RHS = (a + b)(a2 β ab + b2)
Using the Distributive Property of Multiplication, we have
RHS = a(a2 β ab + b2)) + b(a2 β ab + b2)
β RHS = a3 β a2b + ab2 + a2b β ab2 + b3
β RHS = a3 β a2b + a2b + ab2 β ab2 + b3
β RHS = a3 β 0 + 0 + b3
β RHS = a3 + b3
β RHS = LHS
Hence proved.
(a + b)3 Formula
(a + b)3 formula is the formula that is used to expand the cube of sum of two numbers and the formula for the same is,
(a + b)3 = a3 + b3 + 3ab(a + b)
a3 + b3 vs a3 β b3
Both formulas of a3 + b3 and a3 β b3 deal with cubes, but in different ways. The key differences between the a3 + b3 formula and the a3 β b3 formula are listed in the following table:
Formula | Expression | Other Name | Explanation |
---|---|---|---|
a3 + b3 | (a + b)(a2 β ab + b2) | Sum of Cubes | Represents the sum of cubes of two terms as |
a3 β b3 | (a β b)(a2 + ab + b2) | Difference of Cubes | Represents the difference of cubes of two terms |
Read More,
Examples on Sum of Cubes Formula
Example 1: Factorize 343a3 + 216 using sum of cubes.
Solution:
Given,
343a3 + 216 = (7a)3 + (6)3
Since, a3 + b3 = (a + b)(a2 β ab + b2)
β 343a3 + 216 = (7a + 6)[(7a)2 β (7a)(6) + (6)2]
β 343a3 + 216 = (7a + 6)[49a2 β 42a + 36]
Example 2: Factorize 8p3 + 27 using sum of cubes.
Solution:
Given,
8p3 + 27 = (2p)3 + (3)3
Since, a3 + b3 = (a + b)(a2 β ab + b2)
β 8p3 + 27 = (2p + 3)[(2p)2 β (2p)(3) + (3)2]
β 8p3 + 27 = (2p + 3)[4p2 β 6p + 9]
Example 3: Factorize 27t3 + 125 using the sum of cubes.
Solution:
Given,
27t3 + 125 = (3t)3 + (5)3
Since, a3 + b3 = (a + b)(a2 β ab + b2)
β 27t3 + 125 = (3t + 5)[(3t)2 β (3t)(5) + (5)2]
β 27t3 + 125 = (3t + 5)[9t2 β 15t + 25]
Example 4: Factorize 64s3 + 125 using sum of cubes.
Solution:
Given,
64s3 + 125 = (4s)3 + (5)3
Since, a3 + b3 = (a + b)(a2 β ab + b2)
β 64s3 + 125 = (4s + 5)[(8s)2 β (4s)(5) + (5)2]
β 64s3 + 125 = (8s + 5)[64s2 β 20s + 25]
Example 5: Factorize 512 + 729v3 using the sum of cubes formula.
Solution:
512 + 729v3 = (8)3 + (9v)3
Since, a3 + b3 = (a + b)(a2 β ab + b2)
β 512 + 729v3 = (8 + 9v)[(8)2 β (8)(9v) + (9v)2]
β 512 + 729v3 = (8 + 9v)[64 β 72v + 729v2]
Practice Problems on a3 + b3 Formula
Q1. Find the factor of 8x3 + 27y3
Q2. Factorize a6 + 64c3
Q3. Facorize a3 + 1331c3
Q4, Find the factors of y3 + 643z3
a3 + b3 Formula: FAQs
1. What is Expansion Formula of a3 + b3?
The expansion formula of a3 + b3 is given as,
a3 + b3 = (a + b)(a2 β ab + b2)
2. What is Difference of Square Formula?
The difference of square formula is the formula in mathematics that is used to find the difference of two squares without actually calculating the squares. The difference of square formula is given as,
a2 β b2 = (a + b)(a β b)
3. What is Use of a3 + b3 Formula?
The a3 + b3 formula is used to find the sum of two cubes without actually calculating the cubes and it helps to simplify algebraic problems. It is also used to solve trigonometric problems and others.
4. What is Sum of First βnβ Cubes?
The sum of first n cubes is given by the formula,
13 + 23 + 33 + β¦ + n3 = n2.(n + 1)2/4
5. What is Sum of Cubes from 1 to 100?
The sum of cubes of 1 to 100 is 6,376,625 which can be represented as:
13 + 23 + 33 + . . . + 1003 = 6,376,625
6. What are Factors of a3 + b3?
The factors of a3 + b3 is given as,
a3 + b3 = (a + b)(a2 + b2 β ab)
7. What is Formula for a3 + b3?
The formula for a3 + b3 is given as, a3 + b3 = (a + b)(a2 β ab + b2)