(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 polynomials. This formula is:
- one of the algebraic identities.
- the formula for the cube of the sum of two terms.
Let us understand (a + b)3 formula in detail in the following section.
What is the (a + b)^3 Formula?
(a + b) whole cube formula says: (a + b)3 = a3 + 3a2b + 3ab2 + b3. To find the cube of a binomial, we will just multiply (a + b)(a + b)(a + b). (a + b)3 formula is also an identity. It holds true for every value of a and b.
Derivation of a plus b whole cube Formula
The (a + b)3 formula can be derived as follows:
(a + b)3 = (a + b)(a + b)(a + b)
= (a2 + 2ab + b2)(a + b) [∵ (a + b)2 = a2 + 2ab + b2]
= a3 + a2b + 2a2b + 2ab2 + ab2 + b3 [∵ By distribution property]
= a3 + 3a2b + 3ab2 + b3 (OR)
= a3 + 3ab (a + b) + b3
Therefore, (a + b)^3 formula is:
(a + b)3 = a3 + 3a2b + 3ab2 + b3
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Examples on (a + b)3 Formula
Example 1: Solve the following expression using suitable algebraic identity: (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 + 8y3 if x + 2y = 6 and xy = 2.
Solution:
To find: x3 + 8y3
Given: x + 2y = 6
xy = 2
Using a plus b whole cube formula,
(a + b)3 = a3 + 3a2b + 3ab2 + b3
Here, a = x; b = 2y
Therefore,
(x + 2y)3 = x3 + 3 × x2 × (2y) + 3 × x × (2y)2 + (2y)3
(x + 2y)3 = x3 + 6x2y + 12xy2 + 8y3
63 = x3 + 6xy(x + 2y) + 8y3
216 = x3 + 6 × 2 × 6 + 8y3
x3 + 8y3 = 144
Answer: x3 + 8y3 = 144
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
FAQ's on (a + b)^3 Formula
What is the Expansion of (a + b)3 Formula?
(a + b)3 formula is read as a plus b whole cube. Its expansion is expressed as (a + b)3 = a3 + 3a2b + 3ab2 + b3
What is the Formula of a Plus b Plus c Whole Cube?
The formula of a plus b plus c whole cube is: (a + b + c)3 = a3 + b3 + c3 + 3 (a + b) (b + c) (c + a).
☛Note: If a + b + c = 0, then we can write a + b = -c, b + c = -a, and c + a = -b. Then the above formula becomes: 03 = a3 + b3 + c3 + 3 (-c) (-a) (-b) ⇒ a3 + b3 + c3 = 3abc. This is also a widely used formula in algebra.
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 plus b whole cube. The (a + b)3 formula is says: (a + b)3 = a3 + 3a2b + 3ab2 + b3
How 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
= 15625
Answer: (20 + 5)3 = 15625.
How To Use the (a + b)3 Formula Give Steps?
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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