If `(4+3sqrt5)/(4-3sqrt5)=a+bsqrt5`,a, b are rational numbers, them (a, b)=

Options

A. `(61/29,(-24)/29)`
B. `((-61)/29,24/29)`
C. `(61/29,24/29)`
D. `((-61)/29,(-24)/29)`

Answer

To solve the problem \(\frac{4 + 3\sqrt{5}}{4 - 3\sqrt{5}} = a + b\sqrt{5}\), where \(a\) and \(b\) are rational numbers, we will rationalize the denominator and express the result in the required form.

### Step-by-Step Solution:

1. **Rationalize the Denominator**:
We multiply the numerator and the denominator by the conjugate of the denominator, which is \(4 + 3\sqrt{5}\):
\[ \frac{4 + 3\sqrt{5}}{4 - 3\sqrt{5}} \cdot \frac{4 + 3\sqrt{5}}{4 + 3\sqrt{5}} = \frac{(4 + 3\sqrt{5})^2}{(4 - 3\sqrt{5})(4 + 3\sqrt{5})} \]

2. **Calculate the Denominator**:
Using the difference of squares formula, we have:
\[ (4 - 3\sqrt{5})(4 + 3\sqrt{5}) = 4^2 - (3\sqrt{5})^2 = 16 - 45 = -29 \]

3. **Calculate the Numerator**:
Now, we expand the numerator:
\[ (4 + 3\sqrt{5})^2 = 4^2 + 2 \cdot 4 \cdot 3\sqrt{5} + (3\sqrt{5})^2 = 16 + 24\sqrt{5} + 45 = 61 + 24\sqrt{5} \]

4. **Combine the Results**:
Now we can write the expression:
\[ \frac{61 + 24\sqrt{5}}{-29} = \frac{61}{-29} + \frac{24\sqrt{5}}{-29} = -\frac{61}{29} - \frac{24}{29}\sqrt{5} \]

5. **Identify \(a\) and \(b\)**:
From the expression \(-\frac{61}{29} - \frac{24}{29}\sqrt{5}\), we can identify:
\[ a = -\frac{61}{29}, \quad b = -\frac{24}{29} \]

Thus, the values of \(a\) and \(b\) are:
\[ (a, b) = \left(-\frac{61}{29}, -\frac{24}{29}\right) \]