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Let's first solve the given expression:

The expression we are working with is:

\[
\frac{7^{-6}}{7}
\]

Using the laws of exponents, specifically:

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

We can simplify this:

\[
\frac{7^{-6}}{7^1} = 7^{-6 - 1} = 7^{-7}
\]

So, the original expression simplifies to \( 7^{-7} \).

Now, let's review the options:

1. \( \frac{1}{7^{-6}} \) — This is equivalent to \( 7^{6} \), not \( 7^{-7} \).
2. \( 7^{-5} \) — This is not equivalent to \( 7^{-7} \).
3. \( \frac{1}{7^7} \) — This is equivalent to \( 7^{-7} \), as \( \frac{1}{7^7} = 7^{-7} \).
4. \( 7^7 \) — This is not equivalent to \( 7^{-7} \).

The correct equivalent expressions are:  
- \( \frac{1}{7^7} \)

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How do negative exponents work in simplifying expressions?
2. What is the rule for dividing exponents with the same base?
3. How do you convert a negative exponent into a fraction?
4. What happens when you multiply two expressions with exponents?
5. How would this work with variables instead of numbers?

**Tip**: Always remember that a negative exponent moves the base to the denominator of a fraction!

Select all the expressions that are equivalent to 7^-6 / 7.

Understand how to simplify the expression 7^-6 / 7 using the laws of exponents. This step-by-step solution applies the division and negative exponent rules to find equivalent expressions. Suitable for Grades 8-10.

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