Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13

Question

with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(43 ⋅ 7−2)−579 ⋅ 4−13=

Answer 1

To simplify the expression \((4^3 \cdot 7^{-2})^{-5} / (7^9 \cdot 4^{-13})\) while generating equivalent expressions with only positive exponents, we can apply the Properties of Integer Exponents.

1. Apply the exponent to the terms in the numerator: \[ (4^3 \cdot 7^{-2})^{-5} = (4^3)^{-5} \cdot (7^{-2})^{-5} = 4^{-15} \cdot 7^{10} \]

2. Rewrite the expression with the modified numerator: \[ = \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} \]

3. Simplify the denominator: \[ = \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} = \frac{4^{-15}}{4^{-13}} \cdot \frac{7^{10}}{7^9} \]

4. Apply the properties of exponents (subtract exponents for the same bases):

   • For \(4\): \[ \frac{4^{-15}}{4^{-13}} = 4^{-15 - (-13)} = 4^{-15 + 13} = 4^{-2} \]
   • For \(7\): \[ \frac{7^{10}}{7^9} = 7^{10 - 9} = 7^{1} \]

5. Combine the results: \[ = 4^{-2} \cdot 7^{1} = \frac{7}{4^2} = \frac{7}{16} \]

Thus, the simplified expression is: \[ \boxed{\frac{7}{16}} \]