To solve the given problem, we need to break it down step by step.
Firstly, we are informed that the power [tex]\( g^2 \)[/tex] is equivalent to 81. This implies:
[tex]\[ g^2 = 81 \][/tex]
To find [tex]\( g \)[/tex], we take the square root of both sides:
[tex]\[ g = \sqrt{81} \][/tex]
This implies:
[tex]\[ g = 9 \quad \text{or} \quad g = -9 \][/tex]
Both [tex]\( g = 9 \)[/tex] and [tex]\( g = -9 \)[/tex] are valid solutions. However, to find the value of [tex]\( 9^{-2} \)[/tex], we can use [tex]\( g = 9 \)[/tex] (since using [tex]\( g = -9 \)[/tex] would lead to the same result as they both correspond to 9 squared positive).
Next, we need to determine [tex]\( 9^{-2} \)[/tex].
By the definition of negative exponents:
[tex]\[ 9^{-2} = \frac{1}{9^2} \][/tex]
We then calculate [tex]\( 9^2 \)[/tex]:
[tex]\[ 9^2 = 81 \][/tex]
Therefore:
[tex]\[ 9^{-2} = \frac{1}{81} \][/tex]
Hence, the correct value is:
[tex]\[\boxed{\frac{1}{81}} \][/tex]
So, [tex]\( 9^{-2} = \frac{1}{81} \)[/tex].
