Question
Solve $x^{2/3}+ x^{1/3} - 2 = 0.$
✓
Answer
Given equation is $x^{2/3}+ x^{1/3} - 2 = 0$
Putting $x^{1/3} = y,$ the given equation becomes
$y^2 + y - 2 = 0$
$\Rightarrow y^2 + 2y - y - 2 = 0$
$\Rightarrow y(y + 2) - 1(y + 2) = 0$
$\Rightarrow (y + 2) (y - 1) = 0$
$\Rightarrow y + 2 = 0$ or $y - 1 = 0$
$\Rightarrow y = -2$ or $y = 1$
But $x^{1/3} = y$
$\therefore x^{1/3} = -2 or x^{1/3} = 1$
$\Rightarrow x =(-2)^3$ or $x = (1)^3$
$\Rightarrow x = -8$ or $x = 1$
Hence, roots are$ -8, 1.$
Putting $x^{1/3} = y,$ the given equation becomes
$y^2 + y - 2 = 0$
$\Rightarrow y^2 + 2y - y - 2 = 0$
$\Rightarrow y(y + 2) - 1(y + 2) = 0$
$\Rightarrow (y + 2) (y - 1) = 0$
$\Rightarrow y + 2 = 0$ or $y - 1 = 0$
$\Rightarrow y = -2$ or $y = 1$
But $x^{1/3} = y$
$\therefore x^{1/3} = -2 or x^{1/3} = 1$
$\Rightarrow x =(-2)^3$ or $x = (1)^3$
$\Rightarrow x = -8$ or $x = 1$
Hence, roots are$ -8, 1.$
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