To determine which expression is equivalent to $(6^7)^{-3}$, let's simplify step by step:

Step 1: Apply the Power Rule

The power rule of exponents states that when you raise a power to another power, you multiply the exponents:

$(a^m)^n = a^{m \cdot n}$

Thus,

$(6^7)^{-3} = 6^{7 \cdot (-3)} = 6^{-21}.$

Step 2: Match the Correct Option

Looking at the options:

1. $6^4 \cdot 6$
2. $\frac{1}{6^4}$
3. $\frac{6^{-8}}{6^{-2}}$
4. $6^{-21}$

The simplified result $6^{-21}$ matches option 4.

Final Answer:

The expression equivalent to $(6^7)^{-3}$ is:

$6^{-21}$ (Option 4).

Would you like me to clarify any step further or solve similar problems? Let me know!

Related Questions:

1. Simplify $(x^5)^{-2}$.
2. Rewrite $\frac{1}{a^3}$ using a negative exponent.
3. Simplify $(2^4)^3$.
4. Determine the value of $6^{-2}$.
5. Simplify $\frac{6^{-5}}{6^{-3}}$.

Tip: When raising a power to another power, always multiply the exponents!