To solve the given problem, let's follow the steps to understand how to compute [tex]\( b^{-2} \)[/tex] where [tex]\( b = 3 \)[/tex].
1. Define the problem: We need to find the value of [tex]\( b^{-2} \)[/tex] when [tex]\( b \)[/tex] is equal to 3.
2. Interpret the exponent: The notation [tex]\( b^{-2} \)[/tex] indicates that we are dealing with a negative exponent. A negative exponent means we are taking the reciprocal of the base raised to the corresponding positive exponent. So, [tex]\( b^{-2} \)[/tex] is equivalent to [tex]\( \frac{1}{b^2} \)[/tex].
3. Substitute the value of [tex]\( b \)[/tex]: Plug the given value of [tex]\( b \)[/tex] (which is 3) into the expression [tex]\( \frac{1}{b^2} \)[/tex].
[tex]\[
b^{-2} = \frac{1}{3^2}
\][/tex]
4. Calculate the square of the base: Compute the square of 3, which is:
[tex]\[
3^2 = 9
\][/tex]
5. Find the reciprocal: Now, take the reciprocal of 9:
[tex]\[
\frac{1}{9}
\][/tex]
6. Conclusion: The value of [tex]\( b^{-2} \)[/tex] when [tex]\( b = 3 \)[/tex] is [tex]\( \frac{1}{9} \)[/tex].
Therefore, the correct answer from the given options is [tex]\( \boxed{\frac{1}{9}} \)[/tex].
