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