Question 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.

Answer 1

To simplify the expression \( (7^{-3})^7 \cdot \frac{7^{20}}{6^{-2}} \) while only using positive exponents, we will apply the properties of integer exponents.

1. Simplify \( (7^{-3})^7 \): \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

2. Rewrite the expression: Now we rewrite the entire expression: \[ 7^{-21} \cdot \frac{7^{20}}{6^{-2}} \]

3. Distributing the division: Using the property of division of exponents, we can rewrite \( \frac{7^{20}}{6^{-2}} \) as: \[ 7^{20} \cdot 6^{2} \]

4. Combine the terms: Now, substitute back into our expression: \[ 7^{-21} \cdot 7^{20} \cdot 6^{2} \]

   Using the property \( a^m \cdot a^n = a^{m+n} \): \[ 7^{-21 + 20} \cdot 6^{2} = 7^{-1} \cdot 6^{2} \]

5. Convert to positive exponents: To express \( 7^{-1} \) with a positive exponent, we use the property \( a^{-n} = \frac{1}{a^n} \): \[ \frac{6^{2}}{7^{1}} = \frac{6^2}{7} \]

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

7. Final expression: Thus, we have: \[ \frac{36}{7} \]

So, the answer as an improper fraction is: \[ \boxed{\frac{36}{7}} \]