If x is a positive integer such than `x^(9)-r and x^(5)=w,` which of the following must be equal to `x^(13)`?

Options

A. `rw-1`
B. `r+w-1`
C. `(r^(2))/(w)`
D. `r^(2)-w`

Answer

To solve the problem, we need to express \( x^{13} \) in terms of \( r \) and \( w \), where \( r = x^9 \) and \( w = x^5 \).

### Step-by-Step Solution:

1. **Understand the given expressions**:
- We have \( r = x^9 \) and \( w = x^5 \).
- We need to find an expression that equals \( x^{13} \).

2. **Express \( x^{13} \) in terms of \( r \) and \( w \)**:
- We can express \( x^{13} \) as follows:
\[ x^{13} = x^{9 + 4} = x^9 \cdot x^4 \]

3. **Substitute \( x^9 \) with \( r \)**:
- From the expression \( x^9 = r \), we can substitute:
\[ x^{13} = r \cdot x^4 \]

4. **Express \( x^4 \) in terms of \( w \)**:
- We know \( w = x^5 \), which means \( x = w^{1/5} \).
- To find \( x^4 \), we can express it as:
\[ x^4 = (w^{1/5})^4 = w^{4/5} \]

5. **Combine the expressions**:
- Now substitute \( x^4 \) back into the expression for \( x^{13} \):
\[ x^{13} = r \cdot w^{4/5} \]

6. **Find the correct option**:
- Now, let's check the options given in the question:
- **Option A**: \( rw - 1 \) does not equal \( x^{13} \).
- **Option B**: \( r + w - 1 \) does not equal \( x^{13} \).
- **Option C**: \( \frac{r^2}{w} \):
\[ \frac{r^2}{w} = \frac{(x^9)^2}{x^5} = \frac{x^{18}}{x^5} = x^{13} \]
This is equal to \( x^{13} \).
- **Option D**: \( r^2 - w \) does not equal \( x^{13} \).

Thus, the correct answer is **Option C: \( \frac{r^2}{w} \)**.