Solve(4 to the 3rd power times 7 to the -2 power) to the power of -5 / 7 to the 9th power times 4 to the power of -13 with only positive exponents, then solve the expression. Your answer will be a simplified expression with no remaining exponents.

Answer 1

To solve the expression $(4^3\cdot 7^{-2}{)}^{-5}÷(7^9\cdot 4^{-13})$ with positive exponents, we will simplify step by step.

1. Start with the expression: $$(4^3\cdot 7^{-2}{)}^{-5}÷(7^9\cdot 4^{-13})$$

2. Apply the negative exponent to the first part of the expression: $$(4^3{)}^{-5}\cdot (7^{-2}{)}^{-5}=4^{-15}\cdot 7^{10}$$

   Thus, we rewrite the expression as: $$4^{-15}\cdot 7^{10}÷(7^9\cdot 4^{-13})$$

3. Rewrite the division as multiplication by the reciprocal: $$=4^{-15}\cdot 7^{10}\cdot \frac{1}{7^9\cdot 4^{-13}}=4^{-15}\cdot 7^{10}\cdot 7^{-9}\cdot 4^{13}$$

4. Combine the exponents:

   • For the base $4$: $$4^{-15}\cdot 4^{13}=4^{-15+13}=4^{-2}$$

   • For the base $7$: $$7^{10}\cdot 7^{-9}=7^{10-9}=7^1$$

   Combining these, we have: $$4^{-2}\cdot 7^1$$

5. Convert to positive exponents: $$=\frac{7}{4^2}=\frac{7}{16}$$

Thus, the final answer, simplified without any remaining exponents, is: $$\frac{7}{16}$$