Solution:

Step 1:

Apply the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$ to move $(12)^{-5}$ to the numerator, resulting in $(12^5 \cdot 12^5)^0$.

Step 2:

Combine the like bases by adding their exponents.

Step 2.1:

Utilize the property of exponents: $a^m \cdot a^n = a^{m+n}$ to sum the exponents of the same base.

$(12^{5+5})^0$

Step 2.2:

Perform the addition of the exponents: $5 + 5 = 10$.

$(12^{10})^0$

Step 3:

Simplify the expression by dealing with the exponent outside the parentheses.

Step 3.1:

Invoke the rule for powers of powers: $(a^m)^n = a^{mn}$ to multiply the exponents.

$12^{10 \cdot 0}$

Step 3.2:

Carry out the multiplication of the exponents: $10 \times 0 = 0$.

$12^0$

Step 4:

Recognize that any number raised to the power of $0$ equals $1$.

$1$

Knowledge Notes:

1. Negative Exponent Rule: The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$. This means that a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.

2. Multiplying Powers with the Same Base: When multiplying powers with the same base, you add the exponents. The formula is $a^m \cdot a^n = a^{m+n}$.

3. Power of a Power Rule: When raising a power to another power, you multiply the exponents. The formula is $(a^m)^n = a^{mn}$.

4. Zero Exponent Rule: The zero exponent rule states that any non-zero base raised to the power of zero is equal to one, $a^0 = 1$, where $a \neq 0$.

5. Simplification Process: The simplification process involves applying the rules of exponents systematically to reduce an expression to its simplest form. This often involves moving terms, combining like terms, and reducing expressions using the properties of exponents.