To solve this problem, we need to determine which of the given expressions are equivalent to $8^4 \cdot 6^4$.

Step 1: Simplify the original expression

Given expression: $8^4 \cdot 6^4$

We can factor out the common exponent: $(8 \cdot 6)^4 = 48^4$ So, $8^4 \cdot 6^4 = 48^4$.

Step 2: Analyze each option to see if it is equivalent to $48^4$.

1. Option 1: $48^{16}$

   • This expression represents $(48^4)^4 = 48^{16}$, which is not equivalent to $48^4$.

2. Option 2: $\frac{48^5}{48}$

   • Simplifying this expression: $\frac{48^5}{48} = 48^{5-1} = 48^4$
   • This is equivalent to $48^4$.

3. Option 3: $\frac{1}{48^4}$

   • This is the reciprocal of $48^4$ and is not equivalent to $48^4$.

4. Option 4: $48^8$

   • This expression represents $(48^4)^2 = 48^8$, which is not equivalent to $48^4$.

Conclusion:

The only expression equivalent to $48^4$ is $\frac{48^5}{48}$.

Analysis of the Selected Answers:

• 48^16: Incorrect (as analyzed).
• $\frac{48^5}{48}$: Correct.
• $\frac{1}{48^4}$: Incorrect.
• 48^8: Incorrect.

It seems there was a mistake in selecting the correct answers.

Would you like more details or have any further questions?

Related Questions:

1. How can you factor common exponents in an expression like $8^4 \cdot 6^4$?
2. What are the properties of exponents that allow simplification like $\frac{48^5}{48} = 48^4$?
3. How can you identify equivalent expressions in multiple-choice questions involving exponents?
4. What is the significance of the base in exponential expressions when simplifying?
5. Can you explain the process of verifying if two expressions are equivalent in a step-by-step manner?

Tip:

Always double-check your work by simplifying each option individually to ensure that you correctly identify equivalent expressions.