Question

Apply the properties of integer exponents to generate equivalent expressions to (4^3 x 7^-2)^-5/7^9 x 4^13 with only positive exponents, then solve the expression. your answer will be a simplified fraction with no remaining exponents.

Answer 1

thanks!

Answer 2

To generate equivalent expressions with only positive exponents, we can utilize the following properties of integer exponents:

1. Product of powers property: (a^m * a^n) = a^(m+n)
2. Quotient of powers property: (a^m / a^n) = a^(m-n)
3. Power of a power property: (a^m)^n = a^(m*n)
4. Negative exponent property: a^(-n) = 1 / a^n
5. Zero exponent property: a^0 = 1

Let's apply these properties to the given expression step by step:

(4^3 * 7^-2)^(-5) / 7^9 * 4^13

Step 1: Apply power of a power property
((4^3)^(-5) * (7^-2)^(-5)) / 7^9 * 4^13

Step 2: Simplify the exponents inside parentheses
(4^(-15) * 7^10) / 7^9 * 4^13

Step 3: Apply the quotient of powers property
4^(-15) * 7^(10-9) * 4^13

Step 4: Simplify the exponents
4^(-15) * 7^1 * 4^13

Step 5: Apply the product of powers property
4^(-15 + 13) * 7^1

Step 6: Simplify the exponent
4^(-2) * 7

Step 7: Apply the negative exponent property
1 / 4^2 * 7

Step 8: Calculate the values
1/16 * 7 = 7/16

Therefore, the final result is 7/16.

Answer 3

You're welcome! If you have any more questions, feel free to ask.