Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (7^-3)^7 x 7^20/6^-2 with only positive exponent, then solve the expression. Provide the answer as an improper fraction with no exponents or decimals.

Answer 1

To simplify the expression \((7^{-3})^7 \times 7^{20} / 6^{-2}\), we will start by applying the properties of integer exponents.

1. **Apply the power of a power property**:
\((a^m)^n = a^{m \cdot n}\)
\[
(7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21}
\]

2. **Combine the expressions**:
\[
7^{-21} \times 7^{20} = 7^{-21 + 20} = 7^{-1}
\]

3. **Then handle the division by \(6^{-2}\)**:
The division can be rewritten as multiplication by the reciprocal:
\[
\frac{7^{-1}}{6^{-2}} = 7^{-1} \times 6^{2}
\]

4. **Convert \(7^{-1}\) to positive exponents**:
\[
7^{-1} = \frac{1}{7}
\]
Therefore, we now have:
\[
\frac{1}{7} \times 6^{2} = \frac{6^2}{7}
\]

5. **Calculate \(6^2\)**:
\[
6^2 = 36
\]
So our expression becomes:
\[
\frac{36}{7}
\]

The final answer, expressed as an improper fraction with no exponents or decimals, is:
\[
\boxed{\frac{36}{7}}
\]