vereinfachen 1/((b-c)^2)+1/((c-a)^2)+1/((a-b)^2)

Lösung

vereinfachen

\frac{1}{( b - c ) ^{2}} + \frac{1}{( c - a ) ^{2}} + \frac{1}{( a - b ) ^{2}}

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

Schritte zur Lösung

\frac{1}{( b - c ) ^{2}} + \frac{1}{( c - a ) ^{2}} + \frac{1}{( a - b ) ^{2}}

kleinstes gemeinsames Vielfache von

(b - c)^{2}, (c - a)^{2}, (a - b)^{2} : (b - c)^{2} (c - a)^{2} (a - b)^{2}

Passe die Brüche mit Hilfe des kgV an

= \frac{( c - a ) ^{2} ( a - b ) ^{2}}{( b - c ) ^{2} ( c - a ) ^{2} ( a - b ) ^{2}} + \frac{( b - c ) ^{2} ( a - b ) ^{2}}{( b - c ) ^{2} ( c - a ) ^{2} ( a - b ) ^{2}} + \frac{( b - c ) ^{2} ( c - a ) ^{2}}{( b - c ) ^{2} ( c - a ) ^{2} ( a - b ) ^{2}}

Wende Bruchregel an:

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}

= \frac{( c - a ) ^{2} ( a - b ) ^{2} + ( b - c ) ^{2} ( a - b ) ^{2} + ( b - c ) ^{2} ( c - a ) ^{2}}{( b - c ) ^{2} ( c - a ) ^{2} ( a - b ) ^{2}}

Vereinfache

(c - a)^{2} (a - b)^{2} + (b - c)^{2} (a - b)^{2} + (b - c)^{2} (c - a)^{2} : b^{4} + 3 a^{2} c^{2} + 2 b^{2} c^{2} + b^{2} a^{2} - 2 c a^{3} + a^{4} + 2 a^{2} b^{2} + c^{2} b^{2} - 2 b a^{3} - 2 b^{3} a - 2 b^{3} c - 2 b c^{3} - 2 c^{3} a + c^{4}

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

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