Which expression is equivalent to [tex]\( 9^{-2} \)[/tex]?

A. [tex]\(-81\)[/tex]
B. [tex]\(-18\)[/tex]
C. [tex]\(\frac{1}{81}\)[/tex]
D. [tex]\(\frac{1}{18}\)[/tex]



Answer :

To solve the expression [tex]\( 9^{-2} \)[/tex], let's break it down step by step:

1. Understand Negative Exponents:
A negative exponent means we take the reciprocal of the base raised to the positive of that exponent. So, [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex].

2. Rewrite [tex]\( 9^{-2} \)[/tex]:
Applying the property of negative exponents, we rewrite [tex]\( 9^{-2} \)[/tex] as:
[tex]\[ 9^{-2} = \frac{1}{9^2} \][/tex]

3. Calculate the Positive Exponent:
Now we compute [tex]\( 9^2 \)[/tex]:
[tex]\[ 9^2 = 9 \times 9 = 81 \][/tex]

4. Substitute the Computed Value Back:
Substitute [tex]\( 81 \)[/tex] back into the reciprocal expression:
[tex]\[ 9^{-2} = \frac{1}{9^2} = \frac{1}{81} \][/tex]

Hence, the expression [tex]\( 9^{-2} \)[/tex] is equivalent to [tex]\(\frac{1}{81}\)[/tex].

This means the correct choice from the given options is:
[tex]\[ \boxed{\frac{1}{81}} \][/tex]