If $\alpha$ and $\beta$ are the roots of the equation $8x^2 - 3x + 27 = 0$, then the value of $\left(\frac{\alpha}{\beta}\right)^{1/2} + \left(\frac{\beta}{\alpha}\right)^{1/2}$ is
Step-by-Step Solution
Key Concept: Use algebraic identities to express symmetric functions in terms of elementary symmetric polynomials
Given $a + b = \frac{4}{3}$, we need to find the value of the expression. Using the algebraic identity, $(a+b)^2 = a^2 + 2ab + b^2 = \frac{16}{9}$. We compute $\left(\frac{a}{b}\right)^2 + \left(\frac{b}{a}\right)^2 = \frac{a^4 + b^4}{a^2b^2}$. Through substitution and simplification with the constraint $a + b = \frac{4}{3}$, the answer evaluates to 4.
Correct Answer: 4