1. **State the problem:** Identify which expression correctly represents the product $(-4) \cdot (-4) \cdot (-4) \cdot (-4) \cdot (-4)$ using exponents. 2. **Recall the exponent rule:** Multiplying the same base repeatedly is expressed as $a^n$ where $a$ is the base and $n$ is the number of times it is multiplied. 3. **Apply the rule:** Since $-4$ is multiplied 5 times, the expression is $(-4)^5$. 4. **Check the options:** - $(-4)^5 \cdot 8^4$ includes an extra factor $8^4$, so incorrect. - $(-4)(5) \cdot (8)(4)$ is multiplication, not exponentiation, so incorrect. - $5^{-4} \cdot 4^8$ changes bases and exponents incorrectly. - $20 \cdot 32$ is a product of numbers unrelated to the original expression. 5. **Conclusion:** The correct representation is $(-4)^5 \cdot 8^4$ if the problem includes $8^4$, but since the original expression is only $(-4)$ multiplied 5 times, the correct simplified form is $(-4)^5$ alone. Since the question asks which represents the expression $(-4) \cdot (-4) \cdot (-4) \cdot (-4) \cdot (-4)$, the answer is $(-4)^5$. **Final answer:** $(-4)^5$ (which corresponds to the first option if considering only the $(-4)^5$ part).