Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents
To apply the properties of integer exponents, we can use the following rules:
1. For any nonzero number x, x^0 = 1.
2. For any integers a, b, and any nonzero number x, x^a * x^b = x^(a+b).
3. For any integers a, b, and any nonzero number x, (x^a)^b = x^(a*b).
Let's apply these rules to simplify the expression (43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13):
Step 1: Simplify (43 ⋅ 7^(-2))^(-579)
Applying rule 2, we know that (43 ⋅ 7^(-2))^(-579) = 43^(-579) * (7^(-2))^(-579).
Step 2: Simplify 7^(-2)
Using rule 1, we know that 7^(-2) = 1/(7^2) = 1/49.
Step 3: Simplify (7^(-2))^(-579)
Applying rule 3, we have (7^(-2))^(-579) = 1/(7^2 * (-579)) = 1/(49 * (-579)) = 1/(-28271).
Therefore, (43 ⋅ 7^(-2))^(-579) = 43^(-579) * 1/(-28271) = 1/(43^579 * 28271).
Step 4: Simplify 4^(-13)
Using rule 1, we know that 4^(-13) = 1/(4^13) = 1/67108864.
Finally, the simplified expression becomes:
1/(43^579 * 28271) * 1/67108864
To multiply fractions, we can multiply the numerators together and the denominators together:
1 * 1 / (43^579 * 28271 * 67108864)
Simplifying further, we get the final answer:
1 / (43^579 * 28271 * 67108864)
Therefore, the solution is the fraction:
1 / (43^579 * 28271 * 67108864)