Questions: x^(4 / 3) - 13 x^(2 / 3) + 42 = 0

Solution Steps

To solve the equation $x^{4/3} - 13x^{2/3} + 42 = 0$, we can use a substitution method. Let $y = x^{2/3}$. Then the equation becomes a quadratic in terms of $y$: $y^2 - 13y + 42 = 0$. We can solve this quadratic equation for $y$ and then back-substitute to find $x$.

Let $y = x^{2/3}$. The given equation $x^{4/3} - 13x^{2/3} + 42 = 0$ can be rewritten as: $$y^2 - 13y + 42 = 0$$

Solve the quadratic equation $y^2 - 13y + 42 = 0$ for $y$: $$y = 6 \quad \text{or} \quad y = 7$$

Back-substitute $y = x^{2/3}$ to find $x$: $$x^{2/3} = 6 \quad \Rightarrow \quad x = 6^{3/2} = 14.6969$$ $$x^{2/3} = 7 \quad \Rightarrow \quad x = 7^{3/2} = 18.5203$$

Final Answer