The problem asks to simplify the expressions $a^{12} \cdot a^4$ and $a^8 \cdot a^8$ and determine whether they are equivalent for all values of $a$.

Step 1: Simplify $a^{12} \cdot a^4$

Using the property of exponents: $a^m \cdot a^n = a^{m+n}$ $a^{12} \cdot a^4 = a^{12+4} = a^{16}$

Step 2: Simplify $a^8 \cdot a^8$

Similarly, $a^8 \cdot a^8 = a^{8+8} = a^{16}$

Step 3: Conclusion

Both expressions simplify to $a^{16}$, meaning: $a^{12} \cdot a^4 = a^8 \cdot a^8$ for all values of $a$, except when $a = 0$ (as 0 raised to any power is 0, and division by 0 is undefined).

Thus, the statement is true for all $a \neq 0$.

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Related Questions:

1. How do properties of exponents apply when multiplying terms with the same base?
2. Can the result change if $a = 0$?
3. What happens if we deal with negative exponents in similar problems?
4. How would this change if one of the exponents was a fraction?
5. What is the general rule for adding exponents?

Tip: Always remember that when multiplying powers with the same base, you simply add the exponents.