2) \( x^{2 / 3}-2 x^{1 / 3}-3=0 \)

To solve the equation \( x^{2/3} - 2x^{1/3} - 3 = 0 \), let's make a substitution. Let \( y = x^{1/3} \). Then, the equation transforms into \( y^2 - 2y - 3 = 0 \). This quadratic can be factored or solved using the quadratic formula. Factoring gives us \( (y - 3)(y + 1) = 0 \), leading to solutions \( y = 3 \) and \( y = -1 \). Reverting back to \( x \), we have \( x^{1/3} = 3 \) resulting in \( x = 27 \), and \( x^{1/3} = -1 \) giving \( x = -1 \). Thus, the solutions are \( x = 27 \) and \( x = -1 \). Now, let's check the solutions to ensure they're valid in the original equation. Plugging \( x = 27 \) back in yields \( 27^{2/3} - 2(27^{1/3}) - 3 = 9 - 6 - 3 = 0 \), confirming that it's correct. Next, for \( x = -1 \): \( (-1)^{2/3} - 2(-1)^{1/3} - 3 = 1 + 2 - 3 = 0 \), which also checks out. Hence, both solutions are valid!