To simplify the expression \(\frac{x^{-9}}{x^{-12}}\), we need to apply the properties of exponents. Specifically, we use the rule:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
In this case, our base \(a\) is \(x\), and our exponents are \(-9\) and \(-12\). Thus, we can rewrite the expression as:
[tex]\[
\frac{x^{-9}}{x^{-12}} = x^{-9 - (-12)}
\][/tex]
Next, we need to simplify the exponent. Subtracting \(-12\) is the same as adding 12:
[tex]\[
-9 - (-12) = -9 + 12
\][/tex]
So, we perform the arithmetic:
[tex]\[
-9 + 12 = 3
\][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
x^3
\][/tex]
So, the final result is:
[tex]\[
\frac{x^{-9}}{x^{-12}} = x^3
\][/tex]
