Answer
$x = 27$ or $x = -8$
Work Step by Step
We can re-write the equation in the following manner using the rules of exponents:
$$x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$$
$$(x^{\frac{1}{3}})^{2} - (x^{\frac{1}{3}}) - 6 = 0$$
$$b^{2} - b - 6 = 0$$ where $b = x^{\frac{1}{3}}$. Since 2 factors of 6 that, when substracted give 1 are 2 and 3, we can factorize our new equation as so:
$$0 = (b-3)(b+2)$$
and solve for $b$:
$b - 3 = 0$ or $b+2 = 0$
$b = 0+3$ or $b=0-2$
$b = 3$ or $b=-2$
Since we had established that $b = x^{\frac{1}{3}}$, we can now solve for $x$:
$x^{\frac{1}{3}} = 3$ or $x^{\frac{1}{3}} = -2$
$(x^{\frac{1}{3}})^{3} = 3^{3}$ or $(x^{\frac{1}{3}})^{3} = (-2)^{3}$
$x = 27$ or $x = -8$
where, if we substitute the values into the original equation, we conclude that both satisfy the original conditions.
