If `alpha` and `beta` are the roots of `4x^2+3x+7=0`, the value of `1/alpha^3+1/beta^3` is

Answer

To solve the problem, we need to find the value of \( \frac{1}{\alpha^3} + \frac{1}{\beta^3} \) where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 4x^2 + 3x + 7 = 0 \).

### Step 1: Identify the coefficients
The quadratic equation is in the form \( ax^2 + bx + c = 0 \).
Here, \( a = 4 \), \( b = 3 \), and \( c = 7 \).

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{3}{4} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{7}{4} \)

### Step 3: Find \( \alpha^3 + \beta^3 \)
We can use the identity:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) \]
Substituting the values we found:
\[ \alpha^3 + \beta^3 = \left(-\frac{3}{4}\right)^3 - 3 \cdot \frac{7}{4} \cdot \left(-\frac{3}{4}\right) \]

### Step 4: Calculate \( \left(-\frac{3}{4}\right)^3 \)
\[ \left(-\frac{3}{4}\right)^3 = -\frac{27}{64} \]

### Step 5: Calculate \( -3 \cdot \frac{7}{4} \cdot \left(-\frac{3}{4}\right) \)
\[ -3 \cdot \frac{7}{4} \cdot \left(-\frac{3}{4}\right) = \frac{63}{16} \]

### Step 6: Combine the results
Now, we combine the two results:
\[ \alpha^3 + \beta^3 = -\frac{27}{64} + \frac{63}{16} \]
To add these fractions, we need a common denominator, which is 64:
\[ \frac{63}{16} = \frac{63 \times 4}{16 \times 4} = \frac{252}{64} \]
Thus,
\[ \alpha^3 + \beta^3 = -\frac{27}{64} + \frac{252}{64} = \frac{225}{64} \]

### Step 7: Calculate \( \alpha^3 \beta^3 \)
Now, we need \( \alpha^3 \beta^3 \):
\[ \alpha^3 \beta^3 = (\alpha \beta)^3 = \left(\frac{7}{4}\right)^3 = \frac{343}{64} \]

### Step 8: Find \( \frac{1}{\alpha^3} + \frac{1}{\beta^3} \)
Using the formula:
\[ \frac{1}{\alpha^3} + \frac{1}{\beta^3} = \frac{\alpha^3 + \beta^3}{\alpha^3 \beta^3} \]
Substituting the values:
\[ \frac{1}{\alpha^3} + \frac{1}{\beta^3} = \frac{\frac{225}{64}}{\frac{343}{64}} = \frac{225}{343} \]

### Final Answer
Thus, the value of \( \frac{1}{\alpha^3} + \frac{1}{\beta^3} \) is:
\[ \frac{225}{343} \]