Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3⋅27^−7)^0/3^4 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

Answer 1

To apply the properties of integer exponents, we need to remember the rules:
1. Product of powers: (a^m) * (a^n) = a^(m+n)
2. Quotient of powers: (a^m) / (a^n) = a^(m-n)
3. Power of a power: (a^m)^n = a^(m*n)
4. Power of a product: (ab)^n = a^n * b^n
5. Power of a quotient: (a/b)^n = a^n / b^n

Using these rules, we can simplify the expression:

(14^3 * 27^(-7))^0 / 3^4

First, let's simplify the exponents of 14 and 27:

14^3 = 14 * 14 * 14 = 2744

27^(-7) = 1/27^7 = 1/(3^(-3))^7 = 1/3^(-21) = 3^21

Now, we can substitute these values back into the expression:

(2744 * 3^21)^0 / 3^4

Anything raised to the power of 0 is equal to 1, so we have:

1 / 3^4

And 3^4 is equal to 81, so the final simplified expression is:

1 / 81

Therefore, the solution is 1/81.

Answer 2

Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)