**Simplify.**
*Rewrite the expression in the form \[9^n\].*

\[\dfrac{9^{-3}}{9^{12}}=\]

To simplify the expression \(\dfrac{9^{-3}}{9^{12}}\) and rewrite it in the form \(9^n\), we can use the properties of exponents.

### Step 1: Apply the Quotient Rule for Exponents
The quotient rule for exponents states that:

\[
\frac{a^m}{a^n} = a^{m-n}
\]

Using this rule:

\[
\dfrac{9^{-3}}{9^{12}} = 9^{-3-12}
\]

### Step 2: Simplify the Exponent
Simplify the exponent by performing the subtraction:

\[
9^{-3-12} = 9^{-15}
\]

### Final Answer
The simplified expression is:

\[
\boxed{9^{-15}}
\]

Would you like further details on how the exponent rules work, or do you have any other questions?

Here are 5 related questions:
1. How would you simplify \(\frac{9^5}{9^2}\)?
2. What is the value of \(9^{-15}\) as a decimal?
3. How do you simplify \(\frac{a^m}{a^n}\) if \(m = n\)?
4. What happens to the exponent when the base is multiplied, i.e., \((a^m)(a^n)\)?
5. Can you explain why \(9^{-3}\) represents a fraction?

**Tip:** Remember that negative exponents indicate reciprocals. \(a^{-n} = \frac{1}{a^n}\).

Learn how to simplify the expression \( \frac{9^{-3}}{9^{12}} \) using exponent rules and rewrite it in the form \( 9^n \). This problem involves applying the quotient rule for exponents to simplify the expression step-by-step.

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