Q. What is the formula of (a + b)³?
Ans:– (a + b)³ is the algebraic expression, which describes the cube of the sum of two numbers.
The derivation of the formula:
The formula of(a + b)³ is:
(a + b)³ = a³+ b³ + 3ab (a + b)
The derivation of this formula is:
(a + b)³ = (a + b)². (a + b)
[Putting the value of the identity;
(a + b)² = a² + b² + 2ab]
(a + b)³ = (a² + b² + 2ab). (a + b)
[Solving the brackets,]
(a + b)³ = a(a² + b² + 2ab) + b(a² + b² + 2ab)
[Performing computation,]
(a + b)³ = a³ + ab² + 2a²b + a²b + b³ + 2ab²)
(a + b)³ = a³ + 3ab² + 3a²b + b³
(a + b)³ = a³ + b³ + 3ab² + 3a²b
The proof of the formula:
For proofing, the formula, take a = 2 and b = 4.
So,
Put a = 2
b = 4
The formula of (a + b)3 is:
(a + b)³ = a³ + b³ + 3ab (a + b)
The proof of this formula is:
(2 + 4)³ = (2 + 4)². (2 + 4)
(2 + 4)³ = (22 + 42 + 2.2.4). (2 + 4)
(2 + 4)³ = 2(22 + 42 + 2.2.4) + 4(22 + 42 + 2.2.4)
(2 + 4)³ = 23 + 2.42 + 2.224 + 224 + 43 + 2.242)
(2 + 4)³= 8 + 32 + 32 + 16 + 64 + 64
(6)³ = 8 + 64 + 16 + 128
216 = 152 + 144
216 = 216
L.H.S = R. H. S
Hence proved.
Other important identities:
- (a – b)³ = a³ – b³ + 3ab² – 3a²b
- (a + b)4 = a4 + 4a³b + 6a²b² + 4ab³ + b4
- (a – b)4 = a4 – 4a³b + 6a²b² – 4ab³ + b4
- a5 – b5 = (a – b)(a4 + a³b + a²b² + ab³ + b4)