If `(3sqrt(3)+5)^(7)=P+F`, where P is an integer and F is a proper fraction, then `F.(P+F)` is equal to

Options

A. `3^(7)`
B. `2^(6)`
C. `3^(6)`
D. `2^(7)`

Answer

To solve the problem, we need to evaluate the expression \((3\sqrt{3} + 5)^7 = P + F\), where \(P\) is an integer and \(F\) is a proper fraction. We will find \(F \cdot (P + F)\).

### Step-by-Step Solution:

1. **Identify the Conjugate**:
We consider the conjugate of \(3\sqrt{3} + 5\), which is \(3\sqrt{3} - 5\). Since the exponent is odd, we can use this property to help us find \(F\).

2. **Set Up the Equations**:
Let:
\[ x = 3\sqrt{3} + 5 \]
\[ y = 3\sqrt{3} - 5 \]
Then, we have:
\[ x^7 + y^7 = (3\sqrt{3} + 5)^7 + (3\sqrt{3} - 5)^7 \]

3. **Calculate \(x^7 + y^7\)**:
Using the binomial theorem, we can expand both \(x^7\) and \(y^7\). However, we are particularly interested in the integer part \(P\) and the fractional part \(F\). The term \(y^7\) will yield a negative value since \(y < 0\).

4. **Use the Identity**:
The identity \(x^7 + y^7 = (x + y)(x^6 - x^5y + x^4y^2 - x^3y^3 + x^2y^4 - xy^5 + y^6)\) can be applied here. However, we can also directly calculate \(x^7\) and \(y^7\) to find \(P\) and \(F\).

5. **Calculate \(x + y\)**:
\[ x + y = (3\sqrt{3} + 5) + (3\sqrt{3} - 5) = 6\sqrt{3} \]

6. **Calculate \(xy\)**:
\[ xy = (3\sqrt{3} + 5)(3\sqrt{3} - 5) = (3\sqrt{3})^2 - 5^2 = 27 - 25 = 2 \]

7. **Calculate \(x^7 + y^7\)**:
We can use the fact that \(y^7\) will be a small negative number, and thus \(x^7\) will dominate. The integer part \(P\) will be derived from \(x^7\).

8. **Find \(F\)**:
Since \(F\) is the fractional part of \(x^7\), we can express \(F\) as:
\[ F = x^7 - P \]

9. **Calculate \(F \cdot (P + F)\)**:
\[ F \cdot (P + F) = F \cdot x^7 \]
Since \(x^7\) is a large number and \(F\) is a small fraction, this product will yield a specific value.

10. **Final Calculation**:
We can conclude that:
\[ F \cdot (P + F) = F \cdot x^7 \]
After calculating \(x^7\) and \(F\), we can find the exact value.