a^2 + a*b + b^2 and c^2 + c*d + d^2 … Part 3

$(a^2 + a \, b + b^2)^2 \; + \; (c^2 + c \, d + d^2)^2$

$= a^4 + 2 \, a^3 \, b + 3 \, a^2 \, b^2 + 2 \, a \, b^3 + b^4 + c^4 + 2 \, c^3 \, d + 3 \, c^2 \, d^2 + 2 \, c \, d^3 + d^4$

$= (a^2 + 2 \, a \, b + a \, c + a \, d + b^2 + b \, c + b \, d + c^2 + 2 \, c \, d + d^2)^2$

$= ((a+b)^2 \; + \; (a+b) \,(c+d) \; + \; (c+d)^2)^2$

$a^4 \; + \; b^4 \; + \; (a+b)^4$
$= \; 2 \, a^4 \; + \; 4 \, a^3 \, b \; + \; 6 \, a^2 \, b^2 \; + \; 4 \, a \, b^3 \; + \; 2 \, b^4$
$= \; 2 \, (a^2 \; + \; a \, b \; + \; b^2)^2$

$a^4 \; + \; b^4 \; + \; (a+b)^4 \; = \; 2 \,(a^2 \; + \; a \,b \; + \; b^2)^2$

$a^4 \; + \; b^4 \; + \; (a+b)^4 \; + \; c^4 \; + \; d^4 \; + \; (c+d)^4$
$= \; 2 \,(a^2+ a b + b^2)^2 \; + \; 2 \,(c^2 + c \, d + d^2)^2$

$(a+b)^4 \; + \; (c+d)^4 \; + \; (a+b+c+d)^4$
$= \; 2 \, (a^2 + 2 \, a \, b + a \, c + a \, d + b^2 + b \, c + b \, d + c^2 + 2 \, c \, d + d^2)^2$
$= \; 2 \,(a^2+ a b + b^2)^2 \; + \; 2 \,(c^2 + c \, d + d^2)^2$

then,

$(a+b)^4 \; + \; (c+d)^4 \; + \; (a+b+c+d)^4$
$= \; a^4 \; + \; b^4 \; + \; (a+b)^4 \; + \; c^4 \; + \; d^4 \; + \; (c+d)^4$

$(a+b+c+d)^4 \; = \; a^4 \; + \; b^4 \; + \; c^4 \; + \; d^4$

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About benvitalis

math grad - Interest: Number theory
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