Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (7−3)7 ⋅ 7206−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)

Answer 1

To rewrite the expression with only positive exponents, we can use the properties of integer exponents:

1) Any number raised to the power of 0 is equal to 1: a^0 = 1

2) When multiplying two numbers with the same base, we add the exponents: a^m * a^n = a^(m+n)

3) When dividing two numbers with the same base, we subtract the exponents: a^m / a^n = a^(m-n)

Let's breakdown the given expression step by step:

Step 1: (7^(-3)) * 7 * (7^(2)) * (20^6 * 2^(-2))

Step 2: (7^(-3)) * 7 * (7^(2)) * (20^6 / 2^2)

Step 3: (1/7^3) * 7 * (7^2) * (20^6 / 2^2)

Step 4: (1/7^3) * 7 * (7^2) * (20^6 / 2^2)

Step 5: (1/7^3) * 7 * (7^2) * (20^6 / 2^2)

Step 6: (1/7^3) * 7 * (7^2) * (20^6 / 2^2)

Step 7: (1/7^3) * 7 * 7^2 * 20^6 / 2^3

Step 8: (1/7^3) * 7^3 * (20^6 / 2^3)

Step 9: (1/7^3) * 7^3 * (10^3)^6 / 2^3

Step 10: (1/7^3) * 7^3 * 10^(3*6) / 2^3

Step 11: (1/7^3) * 7^3 * 10^18 / 2^3

Now, let's simplify the expression:

Step 12: (1/7^3) * 7^3 * 10^18 / 2^3 = 1 * 7^3 * 10^18 / 2^3

Step 13: 7^3 * 10^18 / 2^3 = (7 * 7 * 7 * 10^18) / (2 * 2 * 2)

Step 14: (7 * 7 * 7 * 10^18) / (2 * 2 * 2) = (343 * 10^18) / 8

Step 15: (343 * 10^18) / 8 = 3430^18 / 8

Step 16: 3430^18 / 8 = (3430^18) / (2^3)

Step 17: (3430^18) / (2^3) = 3430^18 / 8 = 813025390625

Therefore, the simplified expression is 813025390625.