For a non-zero rational number `a, a^(7) div a^(-12)` is equal to

Options

A. `a^5`
B. `a^(-19)`
C. `a^(-5)`
D. `a^(19)`

Answer

To solve the expression \( \frac{a^7}{a^{-12}} \) for a non-zero rational number \( a \), we can use the properties of exponents.

### Step-by-Step Solution:

1. **Identify the expression**: We start with the expression \( \frac{a^7}{a^{-12}} \).

2. **Apply the property of exponents**: When dividing two expressions with the same base, we subtract the exponents. The property states:
\[ \frac{x^m}{x^n} = x^{m-n} \]
Here, our base is \( a \), \( m = 7 \), and \( n = -12 \).

3. **Subtract the exponents**: We apply the property:
\[ \frac{a^7}{a^{-12}} = a^{7 - (-12)} \]

4. **Simplify the exponent**: The subtraction of a negative exponent is equivalent to addition:
\[ 7 - (-12) = 7 + 12 = 19 \]

5. **Write the final result**: Therefore, we have:
\[ a^{7 - (-12)} = a^{19} \]

Thus, the expression \( \frac{a^7}{a^{-12}} \) simplifies to \( a^{19} \).

### Final Answer:
\[ \frac{a^7}{a^{-12}} = a^{19} \]