To solve the expression $\frac{3^{-6}}{3^{-1}}$ using indices (laws of exponents), we can apply the quotient rule of exponents. The quotient rule states that:

$\frac{a^m}{a^n} = a^{m-n}$

In this case, $a = 3$, $m = -6$, and $n = -1$. So,

$\frac{3^{-6}}{3^{-1}} = 3^{-6 - (-1)} = 3^{-6 + 1} = 3^{-5}$

Thus, the simplified expression is:

$3^{-5}$

Would you like further details or any additional questions? Here are five related questions to deepen your understanding:

1. How would you simplify $\frac{5^{-2}}{5^{4}}$?
2. What is the value of $3^{-5}$ as a decimal or fraction?
3. Can you apply the same rule for $\frac{2^7}{2^3}$?
4. How do negative exponents relate to fractions?
5. If $x^{-4} = \frac{1}{x^4}$, how would you express $x^{-6}$ in terms of fractions?

Tip: When dividing exponents with the same base, always subtract the exponents to simplify the expression!