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.(1 point)
(7 ^-3) ^ 7 * 7 ^ 20 / 6 ^-2 = ?

Answer 1

To apply the properties of integer exponents, we can use the following rules:

1. When raising a power to another power, multiply the exponents.
2. When dividing two powers with the same base, subtract the exponents.
3. A negative exponent can be converted to a positive exponent by taking the reciprocal of the base.

Using these rules, we can simplify the expression:

(7^-3)^7 * 7^20 / 6^-2 =

Since (7^-3)^7 = 7^(-3*7) = 7^(-21), and 6^-2 = 1/6^2, we can substitute these values:

7^(-21) * 7^20 / (1/6^2) =

Now, let's simplify further using the rule of multiplying and dividing with the same base:

= 7^(-21 + 20) / (1/6^2)
= 7^(-1) / (1/6^2)
= 1/7^1 * 6^2
= 6^2/7
= 36/7

Therefore, the simplified expression is 36/7.