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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!

Simplify the expression 3^(-6) / 3^(-1) using indices.

Learn how to simplify the expression 3^(-6) / 3^(-1) using the quotient rule of exponents. This example demonstrates how to handle negative exponents and apply exponent rules step-by-step. Suitable for students in Grades 8-10.

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