a^3+b^3 Formula
The a3 + b3 formula is called the sum of cubes (of two numbers) formula. The a cube plus b cube formula is used to find the sum of the two cubes without actually computing the cubes. Also, it is used to factorize the binomials of cubes. Here, we will be discussing the different aspects of the a3 + b3 formula, along with solved examples, and understand where this identity can be applied.
What is a^3+b^3 Formula?
The a3 + b3 formula or the difference of cubes formula is mentioned below:
a3 + b3 = (a + b) (a2 - ab + b2)
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Let us learn the a3 + b3 formula with a few solved examples. The a3 + b3 formula can be verified, by multiplying (a + b) (a2 - ab + b2) and see whether you get a3 + b3.
Proof of a Cube plus b Cube Formula
Let us verify the "a cube plus b cube" formula here. To prove or verify that a3 + b3 = (a + b) (a2 - ab + b2) we need to prove here LHS = RHS.
LHS = a3 + b3
On simplifying the RHS side we get,
= (a + b) (a2 - ab + b2)
On multiplying the a and b separately with (a2 - ab + b2) we get
= 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
Hence, a3 + b3 = (a + b) (a2 - ab + b2).
Examples on a^3 + b^3 Formula
Example 1: Find the value of 1023 + 83 by using the a^3+b^3 formula.
Solution: To find: 1023 + 83.
Let us assume that a = 102 and b = 8.
We will substitute these in a cube plus b cube formula i.e.,
a3 + b3 = (a + b) (a2 - ab + b2)
1023 + 83 = (102 + 8) (1022 - (102)(8) + 82)
= (110) (10404 - 816 + 64)
= (110) (9652)
= 1061720
Answer: 1023 + 83 = 1,061,720.
Example 2: Factorize the expression 8x3 + 27 by using the a^3+b^3 formula.
Solution: To factorize: 8x3 + 27.
We will use the a3 + b3 formula to factorize this.
We can write the given expression as
8x3 + 27 = (2x)3 + 33
We will substitute a = 2x and b = 3 in the formula of a3 + b3.
a3 + b3 = (a + b) (a2 - ab + b2)
(2x)3 + 33 =(2x + 3) ((2x)2 - (2x)(3) + 32)
= (2x + 3) (4x2 - 6x + 9)
Answer: 8x3 + 27 = (2x + 3) (4x2 - 6x + 9).
Example 3: Simplify 193 + 203 using a cube plus b cube formula
Solution: To find 193 + 203
Let us assume a = 19 and b = 20
Using formula a3 + b3 = (a + b) (a2 - ab + b2)
We will substitute these in the a^3 + b^3 formula
a3 + b3 = (a + b) (a2 - ab + b2)
193 + 203 = (19 + 20)(192 - (19)(20) + 202)
= (39) (361 - 380 + 400)
= (39) (381)
= 14,859
Answer: 193 + 203 = 14859.
FAQs on a^3 + b^3 Formula
What is the Expansion of a3 + b3 Formula in Algebra?
a3 + b3 formula is read as a cube plus b cube. It is called the sum of cubes formula. The formula says a3 + b3 = (a + b) (a2 - ab + b2).
How to Apply the a3 + b3 Formula?
Let us understand the application of the a3 + b3 formula with the help of the following example.
Example: Evaluate the value of 183 + 23 using the a3 + b3 formula.
To find: 183 + 23, let us assume that a = 18 and b = 2. We will substitute these in the formula of a3 + b3.
a3 + b3 = (a + b) (a2 - ab + b2)
183 + 23 = (18 + 2)(182 - (18)(2) + 22)
= (20) (324 - 36 + 4)
= 5840
Answer: 183 + 23 = 5840.
What is a Cube Plus b Cube Plus c Cube Formula?
In algebra, we have a formula a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca). By adding 3abc on both sides, we get a3 + b3 + c3 = (a + b + c)(a2 + b2 + c2 - ab - bc - ca) + 3abc.
☛Note: When a + b + c = 0, a3 + b3 + c3 = 3abc from the above formula.
What is a Cube Plus b Cube Formula in Algebra?
The a3 + b3 formula is one of the algebraic identities. It is pronounced as "a cube plus b cube". The a3 + b3 formula is: a3 + b3 = (a + b) (a2 - ab + b2).
How to Use the a^3 + b^3 Formula?
The following steps are followed while using a^3 + b^3 formula.
- Step 1: Express the given expression as sum of cubes.
- Step 2: Recall the formula a3 + b3 = (a + b) (a2 - ab + b2)
- Step 3: Substitute the values of a and b in the a3 + b3 formula and simplify it.
What are the Formulas of a Cube Plus b Cube and a Cube Minus b Cube?
Here are the formulas:
- a3 + b3 = (a + b) (a2 - ab + b2)
- a3 - b3 = (a - b) (a^2 + ab + b^2)
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