If `alpha, beta` are the roots of the equation `8x^(2)-3x+27=0`, then the value of `((alpha^(2))/(beta))^(1//3) +((beta^(2))/(alpha))^(1//3)` is

Options

A. `1//4`
B. `1//3`
C. `7//2`
D. 4

Answer

To solve the problem, we will follow these steps:

1. **Identify the coefficients of the quadratic equation**: The given equation is \(8x^2 - 3x + 27 = 0\). Here, \(a = 8\), \(b = -3\), and \(c = 27\).

2. **Use Vieta's formulas to find the sum and product of the roots**:
- The sum of the roots \(\alpha + \beta\) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-3}{8} = \frac{3}{8} \]
- The product of the roots \(\alpha \beta\) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{27}{8} \]

3. **Set up the expression to evaluate**:
We need to find the value of:
\[ \left(\frac{\alpha^2}{\beta}\right)^{\frac{1}{3}} + \left(\frac{\beta^2}{\alpha}\right)^{\frac{1}{3}} \]

4. **Rewrite the expression with a common denominator**:
We can express the sum as:
\[ \frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} = \frac{\alpha^3 + \beta^3}{\alpha \beta} \]
Thus, we can write:
\[ \left(\frac{\alpha^2}{\beta}\right)^{\frac{1}{3}} + \left(\frac{\beta^2}{\alpha}\right)^{\frac{1}{3}} = \frac{(\alpha^3 + \beta^3)^{\frac{1}{3}}}{(\alpha \beta)^{\frac{1}{3}}} \]

5. **Calculate \(\alpha^3 + \beta^3\)** using the identity:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2) \]
We know:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(\frac{3}{8}\right)^2 - 2\left(\frac{27}{8}\right) = \frac{9}{64} - \frac{54}{8} = \frac{9}{64} - \frac{432}{64} = -\frac{423}{64} \]
Thus,
\[ \alpha^3 + \beta^3 = \left(\frac{3}{8}\right)\left(-\frac{423}{64} + \frac{27}{8}\right) \]

6. **Calculate \(\alpha^3 + \beta^3\)**:
\[ \frac{27}{8} = \frac{216}{64} \]
Therefore,
\[ \alpha^3 + \beta^3 = \frac{3}{8}\left(-\frac{423}{64} + \frac{216}{64}\right) = \frac{3}{8}\left(-\frac{207}{64}\right) = -\frac{621}{512} \]

7. **Substitute back into the expression**:
Now we substitute back:
\[ \left(\frac{-\frac{621}{512}}{\frac{27}{8}}\right)^{\frac{1}{3}} = \left(-\frac{621 \cdot 8}{512 \cdot 27}\right)^{\frac{1}{3}} = \left(-\frac{4968}{13824}\right)^{\frac{1}{3}} = -\frac{1}{4} \]

8. **Final answer**:
The value of the expression is:
\[ \frac{3}{2} \]