Apply the Properties of Integer Exponents generate equivalent express 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 generate an equivalent expression with only positive exponents, we can use the property $(a^m)^n = a^{mn}$.

Applying this property to $(7^{-3})^7$, we get $7^{-21}$.

Applying the same property to $\left(\frac{7^{20}}{6^{-2}}\right)$, we get $\frac{7^{20}}{6^{-12}}$.

Now, we can simplify the expression by applying the property $\frac{a^m}{a^n} = a^{m-n}$.

Using this property, we get $\frac{7^{20}}{6^{-12}} = 7^{20-(-12)} = 7^{20+12} = 7^{32}$.

Therefore, the simplified expression is $7^{-21} \cdot 7^{32}$.

Now, we can use the property $a^m \cdot a^n = a^{m+n}$ to multiply $7^{-21} \cdot 7^{32}$.

Using this property, we get $7^{-21} \cdot 7^{32} = 7^{-21+32} = 7^{11}$.

Therefore, the expression simplifies to $7^{11}$.

To express this as an improper fraction, we can write $7^{11}$ as $\frac{7^{11}}{1}$.

Therefore, the answer is $\frac{7^{11}}{1}$.