To solve the given problem, we need to work with the provided expressions and equations involving the variables [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. Let's analyze the information step-by-step:
1. Given equations:
[tex]\[
r^b = r^b
\][/tex]
[tex]\[
y^a = 100
\][/tex]
2. We are given that [tex]\(y = 10^b\)[/tex]. This can be substituted into the second equation:
[tex]\[
(10^b)^a = 100
\][/tex]
3. Simplify the left-hand side of the equation using the properties of exponents:
[tex]\[
10^{a \cdot b} = 100
\][/tex]
4. The right-hand side [tex]\(100\)[/tex] can be rewritten as a power of 10:
[tex]\[
10^{a \cdot b} = 10^2
\][/tex]
5. Given that both sides of the equation are powers of 10, their exponents must be equal for the expression to hold:
[tex]\[
a \cdot b = 2
\][/tex]
Thus, we have proven that the product of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is equal to 2.
```latex
Therefore, the correct answer is that [tex]\(a \cdot b = 2\)[/tex].
```
