If `x**y=x^(2)+y^(2)+1`, what is the value of `(4**5)/(3**2)`?

Options

A. `(3)/(7)`
B. 3
C. `(10)/(3)`
D. 10

Answer

To solve the problem, we need to evaluate the expression \((4^5)/(3^2)\) using the given operation defined as \(x \star y = x^2 + y^2 + 1\).

### Step-by-Step Solution:

1. **Define the operation**: According to the problem, the operation \(x \star y\) is defined as:
\[ x \star y = x^2 + y^2 + 1 \]

2. **Calculate \(4 \star 5\)**:
- Substitute \(x = 4\) and \(y = 5\) into the operation:
\[ 4 \star 5 = 4^2 + 5^2 + 1 \]
- Calculate \(4^2\) and \(5^2\):
\[ 4^2 = 16 \quad \text{and} \quad 5^2 = 25 \]
- Now substitute these values back into the equation:
\[ 4 \star 5 = 16 + 25 + 1 = 42 \]

3. **Calculate \(3 \star 2\)**:
- Substitute \(x = 3\) and \(y = 2\) into the operation:
\[ 3 \star 2 = 3^2 + 2^2 + 1 \]
- Calculate \(3^2\) and \(2^2\):
\[ 3^2 = 9 \quad \text{and} \quad 2^2 = 4 \]
- Now substitute these values back into the equation:
\[ 3 \star 2 = 9 + 4 + 1 = 14 \]

4. **Calculate \(\frac{4 \star 5}{3 \star 2}\)**:
- Now that we have both values, we can compute:
\[ \frac{4 \star 5}{3 \star 2} = \frac{42}{14} \]
- Simplifying this gives:
\[ \frac{42}{14} = 3 \]

### Final Answer:
The value of \(\frac{4 \star 5}{3 \star 2}\) is \(3\).