apply the properties of integer exponents to generate equivalent expressions to (7 to the power of -3)to the power of 7 x 7 to the power of 20 over 6 to the power of -2 with only positive exponents then solve the expression provide the answer as an improper fraction

Answer 1

To simplify the expression \((7^{-3})^7 \times \frac{7^{20}}{6^{-2}}\) using the properties of integer exponents, we will follow these steps:

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. Convert negative exponents to positive: \[ 7^{-1} = \frac{1}{7} \]

4. Handle \(6^{-2}\): \[ 6^{-2} = \frac{1}{6^2} = \frac{1}{36} \]

5. Rewrite the entire expression: Now we have: \[ \frac{1}{7} \div \frac{1}{36} = \frac{1}{7} \times 36 = \frac{36}{7} \]

The final expression in improper fraction form is: \[ \frac{36}{7} \]