If `(3sqrt(3)+5)^n=p+f.` where p is an integer and f is a proper fraction. then find the value of `(3sqrt3-5)^n,nin,N,` is

Options

A. `1-f`, if n is even
B. `1-f`, if n is odd
C. f, if n is odd
D. f, if n is even

Answer

To solve the problem, we need to analyze the expression \((3\sqrt{3} + 5)^n = p + f\), where \(p\) is an integer and \(f\) is a proper fraction. We are tasked with finding the value of \((3\sqrt{3} - 5)^n\) for \(n \in \mathbb{N}\).

### Step 1: Understand the Binomial Expansion
We start with the binomial expansion of \((3\sqrt{3} + 5)^n\):
\[ (3\sqrt{3} + 5)^n = \sum_{k=0}^{n} \binom{n}{k} (3\sqrt{3})^{n-k} (5)^k \]
This expansion will yield both integer and fractional parts.

### Step 2: Analyze the Terms
The first term of the expansion is \(\binom{n}{0} (3\sqrt{3})^n\), and the last term is \(\binom{n}{n} (5)^n\). The terms alternate in sign when we consider \((3\sqrt{3} - 5)^n\).

### Step 3: Consider the Expression for \((3\sqrt{3} - 5)^n\)
Using the binomial theorem again, we can express \((3\sqrt{3} - 5)^n\):
\[ (3\sqrt{3} - 5)^n = \sum_{k=0}^{n} \binom{n}{k} (3\sqrt{3})^{n-k} (-5)^k \]
This will yield a similar structure but with alternating signs for the terms involving \(5\).

### Step 4: Relate the Two Expressions
Now, we can relate the two expressions:
\[ (3\sqrt{3} + 5)^n + (3\sqrt{3} - 5)^n = 2 \sum_{k \text{ even}} \binom{n}{k} (3\sqrt{3})^{n-k} (5)^k \]
This sum includes only the even indexed terms, which will be integers.

### Step 5: Analyze the Fractional Parts
Since \((3\sqrt{3} + 5)^n = p + f\) and \((3\sqrt{3} - 5)^n\) will yield a similar structure, we can denote:
\[ (3\sqrt{3} - 5)^n = g + h \]
where \(g\) is an integer and \(h\) is a proper fraction.

### Step 6: Determine the Relationship
For \(n\) even, we find:
\[ f + h = 1 \quad \text{(since both are proper fractions)} \]
For \(n\) odd, we find:
\[ f = h \]

### Conclusion
Thus, we can summarize:
- If \(n\) is even, then \((3\sqrt{3} - 5)^n = 1 - f\).
- If \(n\) is odd, then \((3\sqrt{3} - 5)^n = f\).

### Final Answer
The value of \((3\sqrt{3} - 5)^n\) is:
- \(1 - f\) if \(n\) is even.
- \(f\) if \(n\) is odd.