solvefor y,x^{2/3}+y^{2/3}=5

Lösung

löse nach

y, x^{\frac{2}{3}} + y^{\frac{2}{3}} = 5

y = 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2} {- 5 \cdot 5^{\frac{1}{2}} < x < 5 \cdot 5^{\frac{1}{2}}}, y = - 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2} {- 5 \cdot 5^{\frac{1}{2}} < x < 5 \cdot 5^{\frac{1}{2}}}

Schritte zur Lösung

x^{\frac{2}{3}} + y^{\frac{2}{3}} = 5

Verschiebe

x^{\frac{2}{3}}

auf die rechte Seite

y^{\frac{2}{3}} = 5 - x^{\frac{2}{3}}

Potenziere beide Seiten der Gleichung mit

3 : y^{2} = 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2}

y^{2} = 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2}

Löse

y^{2} = 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2} : y = 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2}, y = - 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2}

y = 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2}, y = - 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2}

Überprüfe die Lösungen:

y = 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2} {- 5 \cdot 5^{\frac{1}{2}} < x < 5 \cdot 5^{\frac{1}{2}}}, y = - 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2} {- 5 \cdot 5^{\frac{1}{2}} < x < 5 \cdot 5^{\frac{1}{2}}}

Die Lösungen sind

y = 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2} {- 5 \cdot 5^{\frac{1}{2}} < x < 5 \cdot 5^{\frac{1}{2}}}, y = - 125 - 75 x^{\frac{2}{3}} + 15 x^{\frac{4}{3}} - x^{2} {- 5 \cdot 5^{\frac{1}{2}} < x < 5 \cdot 5^{\frac{1}{2}}}

4 w + 4 = - 4

- 2 x^{\frac{3}{4}} + 47 = - 7

x^{\frac{1}{2}} - 3 x^{\frac{1}{4}} + 2 = 0

x - 3 = 1 + x - 4

n - 1 + 2 = 3 n + 3