Identify the $a \text{ \# } b$ Pattern Rule
The question presents a custom mathematical operation '#'. We need to find the rule that defines this operation using the provided examples.
Analyzing Examples
We are given:
- $2 \text{ \# } 3 = 13$
- $3 \text{ \# } 4 = 25$
Let's test the hypothesis that the operation follows the rule $a \text{ \# } b = a^2 + b^2$.
- For the first case ($a=2, b=3$): $2^2 + 3^2 = 4 + 9 = 13$. This matches.
- For the second case ($a=3, b=4$): $3^2 + 4^2 = 9 + 16 = 25$. This also matches.
The pattern $a \text{ \# } b = a^2 + b^2$ is confirmed.
Calculate $4 \text{ \# } 5$
Using the confirmed pattern $a \text{ \# } b = a^2 + b^2$, we can now calculate $4 \text{ \# } 5$.
Here, $a=4$ and $b=5$.
$ 4 \text{ \# } 5 = 4^2 + 5^2 $
Calculate the terms:
- $4^2 = 16$
- $5^2 = 25$
Add the results:
$ 16 + 25 = 41 $
Thus, $4 \text{ \# } 5 = 41$.
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