To find the value of [tex]\( b \)[/tex] in the equation [tex]\( 9^{-8} \cdot 9^{-2} = a^b \)[/tex], let's proceed step-by-step:
1. Applying properties of exponents:
When you multiply terms with the same base, you add the exponents. Therefore,
[tex]\[
9^{-8} \cdot 9^{-2} = 9^{(-8 + -2)}
\][/tex]
2. Simplify the exponent:
Combine the exponents on the right-hand side:
[tex]\[
(-8) + (-2) = -10
\][/tex]
So,
[tex]\[
9^{-8} \cdot 9^{-2} = 9^{-10}
\][/tex]
3. Comparing with [tex]\( a^b \)[/tex]:
Now, [tex]\( 9^{-10} \)[/tex] should be written in the form [tex]\( a^b \)[/tex]. Given [tex]\( a = 9 \)[/tex], we can directly equate the exponents on both sides:
[tex]\[
9^{-10} = 9^b
\][/tex]
4. Finding [tex]\( b \)[/tex]:
By equating the exponents from the bases that are the same, we get:
[tex]\[
b = -10
\][/tex]
Therefore, the value of [tex]\( b \)[/tex] is
\boxed{-10}
