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Beliebt Algebra >

x^{4/3}-5x^{2/3}+6=0

Lösung

x^{\frac{4}{3}} - 5 x^{\frac{2}{3}} + 6 = 0

Lösung

x = 3 \cdot 3^{\frac{1}{2}}, x = - 3 \cdot 3^{\frac{1}{2}}, x = 2 \cdot 2^{\frac{1}{2}}, x = - 2 \cdot 2^{\frac{1}{2}}

+1

Dezimale

x = 5.19615 \dots, x = - 5.19615 \dots, x = 2.82842 \dots, x = - 2.82842 \dots

Schritte zur Lösung

x^{\frac{4}{3}} - 5 x^{\frac{2}{3}} + 6 = 0

Verwende die folgende Exponenteneigenschaft

: a^{\frac{n}{m}} = (a^{\frac{1}{m}})^{n}

x^{\frac{4}{3}} = (x^{\frac{1}{3}})^{4}, x^{\frac{2}{3}} = (x^{\frac{1}{3}})^{2}

(x^{\frac{1}{3}})^{4} - 5 (x^{\frac{1}{3}})^{2} + 6 = 0

Schreibe die Gleichung um mit

x^{\frac{1}{3}} = u

u^{4} - 5 u^{2} + 6 = 0

Löse

u^{4} - 5 u^{2} + 6 = 0 : u = 3, u = - 3, u = 2, u = - 2

u = 3, u = - 3, u = 2, u = - 2

Setze

u = x^{\frac{1}{3}} w i e d er e in,

löse für

x

Löse

x^{\frac{1}{3}} = 3 : x = 3 \cdot 3^{\frac{1}{2}}

Löse

x^{\frac{1}{3}} = - 3 : x = - 3 \cdot 3^{\frac{1}{2}}

Löse

x^{\frac{1}{3}} = 2 : x = 2 \cdot 2^{\frac{1}{2}}

Löse

x^{\frac{1}{3}} = - 2 : x = - 2 \cdot 2^{\frac{1}{2}}

x = 3 \cdot 3^{\frac{1}{2}}, x = - 3 \cdot 3^{\frac{1}{2}}, x = 2 \cdot 2^{\frac{1}{2}}, x = - 2 \cdot 2^{\frac{1}{2}}

Überprüfe die Lösungen:

x = 3 \cdot 3^{\frac{1}{2}}

Wahr

, x = - 3 \cdot 3^{\frac{1}{2}}

Wahr

, x = 2 \cdot 2^{\frac{1}{2}}

Wahr

, x = - 2 \cdot 2^{\frac{1}{2}}

Wahr

Die Lösungen sind

x = 3 \cdot 3^{\frac{1}{2}}, x = - 3 \cdot 3^{\frac{1}{2}}, x = 2 \cdot 2^{\frac{1}{2}}, x = - 2 \cdot 2^{\frac{1}{2}}

Graph

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