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Explicação passo-a-passo:

[tex]x^(0,333...) + x^(0,666...) = 7

0.333... = \frac{1}{3} \\ 0.666... = \frac{2}{3} \\ \\ x {}^{ \frac{1}{3} } + x {}^{ \frac{2}{3} } = 7 \\ x {}^{ \frac{1}{3}} + (x {}^{ \frac{1}{3} } ) {}^{2} = 7 \\ x {}^{ \frac{1}{3} } = y \\ y + y {}^{2} = 7 \\ y {}^{2} + y + \frac{1}{4} = 7 + \frac{1}{4} \\ (y + \frac{1}{4} ) {}^{2} = \frac{29}{4} \\ \sqrt{(y + \frac{1}{2} ) {}^{2} } = \sqrt{ \frac{29}{4} } \\ |y + \frac{1}{2} | = \frac{ \sqrt{29} }{2} \\ y = - \frac{1}{2} - \frac{ \sqrt{29} }{2} \\ ou \\ y = \frac{ \sqrt{29} }{2} - \frac{1}{2} \\ x {}^{ \frac{1}{3} } = y \\ x {}^{ \frac{1}{3} } = \frac{ \sqrt{29} }{2} - \frac{1}{2} \\ (x {}^{ \frac{1}{3} } ) {}^{3} = ( \frac{ \sqrt{29} }{2} - \frac{1}{2} ) {}^{ \frac{1}{3} } \\x = ( \frac{ \sqrt{29} - 1}{2}) {}^{3}[/tex]