To evaluate the expression [tex]\( 6^{-6} \div 6^{-3} \)[/tex], let's examine the properties of exponents and follow step-by-step.
1. Expression Given:
[tex]\[
6^{-6} \div 6^{-3}
\][/tex]
2. Applying the Quotient Rule for exponents:
The quotient rule states that when dividing like bases with exponents, you subtract the exponents. So,
[tex]\[
6^{-6} \div 6^{-3} = 6^{-6 - (-3)} = 6^{-6 + 3}
\][/tex]
3. Simplification of the exponent:
[tex]\[
6^{-6 + 3} = 6^{-3}
\][/tex]
4. Evaluating [tex]\( 6^{-3} \)[/tex]:
Negative exponent indicates the reciprocal of the base raised to the positive exponent.
[tex]\[
6^{-3} = \frac{1}{6^3}
\][/tex]
5. Computing the Power:
[tex]\[
6^3 = 6 \times 6 \times 6 = 216
\][/tex]
6. Putting it Together:
[tex]\[
6^{-3} = \frac{1}{216}
\][/tex]
So, the expression [tex]\( 6^{-6} \div 6^{-3} \)[/tex] evaluates to [tex]\(\frac{1}{216}\)[/tex].
Hence, the correct answer is [tex]\(\frac{1}{216}\)[/tex].
