Apply the Properties of Integer Exponents to generate equivalent expressions to (7^−3)^7 ⋅ 7^20/6^−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point) (7^−3)^7 ⋅ 7^20/6^−2=

Answer 1

To solve the expression \((7^{-3})^7 \cdot 7^{20} / 6^{-2}\) using the properties of integer exponents, we will first simplify each part.

1. Simplify \((7^{-3})^7\): Using 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 terms: Now we substitute back into the expression: \[ 7^{-21} \cdot 7^{20} \] Using the product of powers property, \(a^m \cdot a^n = a^{m+n}\): \[ 7^{-21 + 20} = 7^{-1} \]

3. Now substitute into the entire expression: We have: \[ \frac{7^{-1}}{6^{-2}} \] We can rewrite \(6^{-2}\) in the denominator using the rule \(\frac{1}{a^{-n}} = a^n\): \[ \frac{7^{-1}}{6^{-2}} = 7^{-1} \cdot 6^{2} \]

4. Rewrite \(7^{-1}\): \[ = \frac{6^2}{7^1} = \frac{6^2}{7} \]

5. Calculate \(6^2\): \[ = \frac{36}{7} \]

Thus, the final simplified expression in improper fraction form is: \[ \boxed{\frac{36}{7}} \]