Apply the properties of integer exponents to generate equivalent expressions to (7^-3)^7x7^20/6^-2 with only positive exponents then solve the expression provide the answer as an improper fraction
To generate equivalent expressions with positive exponents, we can apply the property that states: (a^m / b^n) = (a^m * b^(-n)).
Applying this property to the given expression, we have ((7^-3)^7 * 7^20 * 6^2)^-1, since 6^-2 = 1/6^2 = 1/(6^2) = (6^(-2))^-1.
Next, let's simplify the expression: ((7^-3)^7 * 7^20 * 6^2)^-1 = (7^(-3 * 7) * 7^20 * 6^2)^-1 = (7^-21 * 7^20 * 6^2)^-1.
Now, applying another property of exponents, we can add the exponents when multiplying the same base: a^m * a^n = a^(m + n).
Using this property, we rewrite the expression: (7^-21 * 7^20 * 6^2)^-1 = (7^(-21 + 20) * 6^2)^-1 = (7^-1 * 6^2)^-1.
Furthermore, we can evaluate 7^-1 as 1/7 and simplify the expression further: (7^-1 * 6^2)^-1 = (1/7 * 6^2)^-1 = (1/7 * 36)^-1 = (36/7)^-1.
Finally, to find the value of the expression, we take the reciprocal: (36/7)^-1 = 1 / (36/7) = 7/36.
Therefore, the solution to the expression (7^-3)^7 * 7^20 * 6^2 / 6^-2, with positive exponents, is 7/36 as an improper fraction.