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 \((4^3 \cdot 7^{-2})^{-5} \div (7^9 \cdot 4^{-13})\) with positive exponents, we will simplify step by step.

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

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: \[ 4^{-15} \cdot 7^{10} \div (7^9 \cdot 4^{-13}) \]

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: \[ \frac{7}{16} \]