Answer :
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].
```
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].
```
