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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 \(a \neq 0\).

Do you want more details or further clarification?

### 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.

Does a^12 * a^4 = a^8 * a^8 for all values of a? Simplify both sides and determine if the expressions are equivalent.

Learn how to simplify the exponential expressions a^12 * a^4 and a^8 * a^8 using the exponent addition rule. Explore whether these two expressions are equivalent for all values of 'a', including a detailed step-by-step solution. Suitable for students in Grades 8-10.

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