To simplify the expression [tex]\(\frac{(6^{-4})^{-9}}{6^6}\)[/tex], we need to follow the rules of exponents very carefully. Let's break it down step by step.
1. Simplify the numerator [tex]\((6^{-4})^{-9}\)[/tex]:
- When we have an exponent raised to another exponent, we use the power of a power rule [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex].
- Apply this rule: [tex]\((6^{-4})^{-9} = 6^{-4 \cdot (-9)} = 6^{36}\)[/tex].
2. Rewrite the expression with the simplified numerator:
- After simplifying the numerator, our expression becomes: [tex]\(\frac{6^{36}}{6^6}\)[/tex].
3. Simplify the fraction [tex]\(\frac{6^{36}}{6^6}\)[/tex]:
- When dividing numbers with the same base, we subtract the exponents: [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
- Apply this rule: [tex]\(\frac{6^{36}}{6^6} = 6^{36 - 6} = 6^{30}\)[/tex].
The final simplified form of the expression is [tex]\(\boxed{6^{30}}\)[/tex].
