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(5y^4)^3=125y^{24}

Solución

(5 y^{4})^{3} = 125 y^{24}

Solución

y = 1, y = - 1, y = i, y = - i, y = \frac{3}{2} + \frac{1}{2} i, y = - \frac{3}{2} - \frac{1}{2} i, y = - \frac{1}{2} + \frac{3}{2} i, y = \frac{1}{2} - \frac{3}{2} i, y = \frac{1}{2} + \frac{3}{2} i, y = - \frac{1}{2} - \frac{3}{2} i, y = - \frac{3}{2} + \frac{1}{2} i, y = \frac{3}{2} - \frac{1}{2} i, y = 0

Pasos de solución

(5 y^{4})^{3} = 125 y^{24}

Desarrollar

(5 y^{4})^{3} : 125 y^{12}

125 y^{12} = 125 y^{24}

Intercambiar lados

125 y^{24} = 125 y^{12}

Mover

125 y^{12}

al lado izquierdo

125 y^{24} - 125 y^{12} = 0

Re-escribir la ecuación con

u = y^{2}, u^{6} = y^{12}

y

u^{12} = y^{24}

125 u^{12} - 125 u^{6} = 0

Resolver

125 u^{12} - 125 u^{6} = 0 : u = 1, u = - 1, u = \frac{1}{2} + \frac{3}{2} i, u = - \frac{1}{2} - \frac{3}{2} i, u = - \frac{1}{2} + \frac{3}{2} i, u = \frac{1}{2} - \frac{3}{2} i, u = 0

u = 1, u = - 1, u = \frac{1}{2} + \frac{3}{2} i, u = - \frac{1}{2} - \frac{3}{2} i, u = - \frac{1}{2} + \frac{3}{2} i, u = \frac{1}{2} - \frac{3}{2} i, u = 0

Sustituir hacia atrás la

u = y^{2},

resolver para

y

Resolver

y^{2} = 1 : y = 1, y = - 1

Resolver

y^{2} = - 1 : y = i, y = - i

Resolver

y^{2} = \frac{1}{2} + \frac{3}{2} i : y = \frac{3}{2} + \frac{1}{2} i, y = - \frac{3}{2} - \frac{1}{2} i

Resolver

y^{2} = - \frac{1}{2} - \frac{3}{2} i : y = - \frac{1}{2} + \frac{3}{2} i, y = \frac{1}{2} - \frac{3}{2} i

Resolver

y^{2} = - \frac{1}{2} + \frac{3}{2} i : y = \frac{1}{2} + \frac{3}{2} i, y = - \frac{1}{2} - \frac{3}{2} i

Resolver

y^{2} = \frac{1}{2} - \frac{3}{2} i : y = - \frac{3}{2} + \frac{1}{2} i, y = \frac{3}{2} - \frac{1}{2} i

Resolver

y^{2} = 0 : y = 0

Las soluciones son

y = 1, y = - 1, y = i, y = - i, y = \frac{3}{2} + \frac{1}{2} i, y = - \frac{3}{2} - \frac{1}{2} i, y = - \frac{1}{2} + \frac{3}{2} i, y = \frac{1}{2} - \frac{3}{2} i, y = \frac{1}{2} + \frac{3}{2} i, y = - \frac{1}{2} - \frac{3}{2} i, y = - \frac{3}{2} + \frac{1}{2} i, y = \frac{3}{2} - \frac{1}{2} i, y = 0

Ejemplos populares

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5 x^{3} + 2 x^{3} + 3 x^{6} = 39

5p(2p-3)(p+7)=0

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2x^3-14x^2+24x=0

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