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What is the Formula of (a + b)³

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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)