- Identify the negative exponent rule: \(x^{-n} = \frac{1}{x^n}\).
- Apply the rule to rewrite \(9^{-2}\) as \(\frac{1}{9^2}\).
- Evaluate the square in the denominator: \(9^2 = 81\).
- Final value is \(\boxed{\frac{1}{81}}\).
Explanation
- Applying the negative exponent rule
To simplify the expression \(9^{-2}\), we recall the rule for negative exponents, which states that for any non-zero real number \(x\) and a positive integer \(n\), the expression \(x^{-n}\) is equivalent to \(\frac{1}{x^n}\). Applying this rule to our expression, we set \(x = 9\) and \(n = 2\), giving:
\[9^{-2} = \frac{1}{9^2}\]
- Calculating the power in the denominator
Now, we calculate the denominator by squaring the base \(9\):
\[9^2 = 9 \times 9 = 81\]
Substituting this value back into our fraction, we get:
\[\frac{1}{9^2} = \frac{1}{81}\]
- Finalizing the simplification
The fraction \(\frac{1}{81}\) is already in its simplest form because the numerator is 1 and the denominator 81 has no common factors other than 1. Thus, the simplified value of the expression \(9^{-2}\) is \(\frac{1}{81}\).
\[9^{-2} = \boxed{\frac{1}{81}}\]
Examples
The concept of negative exponents is useful in scientific notation and engineering to express very small numbers. For example, if a measurement of a microscopic object is given as \(9^{-2}\) millimeters, it simply means \(\frac{1}{81}\) of a millimeter. This notation allows scientists to handle magnitudes across vast scales using compact mathematical expressions.