# How do you simplify  [ (a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 ] / [ (a^4 - b^4)^3 + (b^4 - c^4)^3 + (c^4 - a^4)^3]?

May 10, 2016

$\frac{{\left({a}^{2} - {b}^{2}\right)}^{3} + {\left({b}^{2} - {c}^{2}\right)}^{3} + {\left({c}^{2} - {a}^{2}\right)}^{3}}{{\left({a}^{4} - {b}^{4}\right)}^{3} + {\left({b}^{4} - {c}^{4}\right)}^{3} + {\left({c}^{4} - {a}^{4}\right)}^{3}} = \frac{1}{\left({a}^{2} + {b}^{2}\right) \left({b}^{2} + {c}^{2}\right) \left({c}^{2} + {a}^{2}\right)}$

excluding any of $a = \pm b$, $b = \pm c$, $c = \pm a$.

#### Explanation:

Notice that:

${\left(A - B\right)}^{3} + {\left(B - C\right)}^{3} + {\left(C - A\right)}^{3}$

$= \left(\textcolor{red}{\cancel{\textcolor{b l a c k}{{A}^{3}}}} - 3 {A}^{2} B + 3 A {B}^{2} - \textcolor{red}{\cancel{\textcolor{b l a c k}{{B}^{3}}}}\right) + \left(\textcolor{red}{\cancel{\textcolor{b l a c k}{{B}^{3}}}} - 3 {B}^{2} C + 3 B {C}^{2} - \textcolor{red}{\cancel{\textcolor{b l a c k}{{C}^{3}}}}\right) + \left(\textcolor{red}{\cancel{\textcolor{b l a c k}{{C}^{3}}}} - 3 {C}^{2} A + 3 C {A}^{2} - \textcolor{red}{\cancel{\textcolor{b l a c k}{{A}^{3}}}}\right)$

$= 3 \left(A {B}^{2} - {A}^{2} B + B {C}^{2} - {B}^{2} C + C {A}^{2} - {C}^{2} A\right)$

$= 3 \left(A - B\right) \left(B - C\right) \left(C - A\right)$

Note also:

${a}^{4} - {b}^{4} = \left({a}^{2} - {b}^{2}\right) \left({a}^{2} + {b}^{2}\right)$, etc.

So:

$\frac{{\left({a}^{2} - {b}^{2}\right)}^{3} + {\left({b}^{2} - {c}^{2}\right)}^{3} + {\left({c}^{2} - {a}^{2}\right)}^{3}}{{\left({a}^{4} - {b}^{4}\right)}^{3} + {\left({b}^{4} - {c}^{4}\right)}^{3} + {\left({c}^{4} - {a}^{4}\right)}^{3}}$

$= \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \left({a}^{2} - {b}^{2}\right) \left({b}^{2} - {c}^{2}\right) \left({c}^{2} - {a}^{2}\right)}{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \left({a}^{4} - {b}^{4}\right) \left({b}^{4} - {c}^{4}\right) \left({c}^{4} - {a}^{4}\right)}$

$= \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left({a}^{2} - {b}^{2}\right)}}} \textcolor{red}{\cancel{\textcolor{b l a c k}{\left({b}^{2} - {c}^{2}\right)}}} \textcolor{red}{\cancel{\textcolor{b l a c k}{\left({c}^{2} - {a}^{2}\right)}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left({a}^{2} - {b}^{2}\right)}}} \left({a}^{2} + {b}^{2}\right) \textcolor{red}{\cancel{\textcolor{b l a c k}{\left({b}^{2} - {c}^{2}\right)}}} \left({b}^{2} + {c}^{2}\right) \textcolor{red}{\cancel{\textcolor{b l a c k}{\left({c}^{2} - {a}^{2}\right)}}} \left({c}^{2} + {a}^{2}\right)}$

$= \frac{1}{\left({a}^{2} + {b}^{2}\right) \left({b}^{2} + {c}^{2}\right) \left({c}^{2} + {a}^{2}\right)}$

excluding any of $a = \pm b$, $b = \pm c$, $c = \pm a$