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To find the numerical equivalent of \( 12^{-5} \times 12^{7} \) using the product rule of integer exponents, we first apply the property of exponents that states:
\[
a^m \times a^n = a^{m+n}
\]

In this equation, \( a \) is the base, and \( m \) and \( n \) are the exponents.

### Step-by-step Calculation:

1. **Identify the base and exponents:**
- The base is \( 12 \).
- The exponents are \( -5 \) and \( 7 \).

2. **Apply the product rule:**
Using the product rule:
\[
12^{-5} \times 12^{7} = 12^{-5 + 7}
\]

3. **Simplify the exponent:**
Now, perform the arithmetic in the exponent:
\[
-5 + 7 = 2
\]

4. **Rewrite the expression:**
Substitute back into the equation:
\[
12^{-5} \times 12^{7} = 12^{2}
\]

5. **Calculate the numerical value:**
Now, we find the numerical equivalent of \( 12^{2} \):
\[
12^{2} = 144
\]

### Conclusion:
Thus, the numerical equivalent of \( 12^{-5} \times 12^{7} \) is \( 144 \).

### Explanation of the Product Rule Usage:
The product rule for exponents allows us to combine terms with the same base by adding their exponents. Since both terms had the base \( 12 \), we were able to simplify the expression by adding \( -5 \) and \( 7 \) to get \( 2 \). This simplification made it possible to express the product as a single exponentiation, which we then calculated to find the final result.

1 answer

To determine the numerical value of \( 12^{-5} \times 12^{7} \) using the product rule for integer exponents, we start by applying the exponent property which states:
\[ a^m \times a^n = a^{m+n} \] Here, \( a \) represents the base, and \( m \) and \( n \) are the exponents.

Step-by-step Calculation:

  1. Identify the base and exponents:

    • The base is \( 12 \).
    • The exponents are \( -5 \) and \( 7 \).

  2. Apply the product rule: Employing the product rule gives us:
    \[ 12^{-5} \times 12^{7} = 12^{-5 + 7} \]

  3. Simplify the exponent: Now, we calculate the sum in the exponent:
    \[ -5 + 7 = 2 \]

  4. Rewrite the expression: Replacing this back into the equation yields:
    \[ 12^{-5} \times 12^{7} = 12^{2} \]

  5. Calculate the numerical value: Finally, we compute the numerical equivalent of \( 12^{2} \):
    \[ 12^{2} = 144 \]

Conclusion:

Therefore, the numerical equivalent of \( 12^{-5} \times 12^{7} \) is \( 144 \).

Explanation of the Product Rule Usage:

The product rule for exponents enables us to merge terms with the same base by summing their exponents. Since both terms had the base \( 12 \), we were able to simplify the expression by adding \( -5 \) and \( 7 \) to obtain \( 2 \). This simplification allowed us to represent the product as a single exponentiation, which we then calculated to arrive at the final answer.