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What is a plus b whole four \( (a+b)^4 \) Formula

There are multiples ways to find out the result of \( (a+b)^4 \) formula.

=> we can write  \( (a+b)^4  = {(a+b)^2}^2\) [we know that \( (a+b)^2 = a^2+b^2+2ab \) ]

\( = (a^2+b^2+2ab)^2 \) [we know that \( (a+b+c)^2 = a^2+b^2+c^2 + 2ab +2bc +2ca \) ]

we need to put the value of \( a = a^2, b= b^2 \; and \; c=2ab \)

\( = (a^2)^2 + (b^2)^2 + (2ab)^2  + 2a^2 b^2 + 2 b^2 \times 2ab + 2 a^2 \times 2ab \)

\( = a^4 + b^4 + 4a^2b^2  + 2a^2 b^2 + 4b^3a + 4a^3b\)

\( = a^4 + b^4 + 6a^2b^2  + 4b^3a + 4a^3b\) [Proof]

Now we are going to proof \( (a+b)^4 \) formula by another method.

we can write \( (a+b)^4  = (a+b)^2 (a+b)^2 \) [we know that \( (a+b)^2 = a^2+b^2+2ab \) ]

\( (a+b)^4  = (a^2+b^2+2ab) (a^2+b^2+2ab) \)

\(  = a^2 (a^2+b^2+2ab) + b^2 (a^2+b^2+2ab) + 2ab (a^2+b^2+2ab) \)

\(  =  a^4+ a^2b^2+2a^3b + b^2 a^2+b^4+2ab^3 + 2a^3b +2ab^3+4a^2b^2 \) [Arrage Simler Value according to power]

\(  =  a^4 +b^4 + a^2b^2+4a^2b^2 + b^2 a^2+2ab^3 + 2ab^3+ 2a^3b +2a^3b\) [add Simler Value]

\(  =  a^4 +b^4 + 6a^2b^2+4ab^3 + 4a^3b \)

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