To simplify the given expression \((4^3 \cdot 7^{-2})^{-5} / (7^9 \cdot 4^{-13})\), we will first apply the properties of integer exponents step by step.
-
Apply the exponent to the numerator: \[ (4^3 \cdot 7^{-2})^{-5} = (4^3)^{-5} \cdot (7^{-2})^{-5} = 4^{-15} \cdot 7^{10} \]
-
Rewrite the expression: Now the expression becomes: \[ \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} \]
-
Apply the laws of exponents to simplify the denominator: We can rewrite the denominator: \[ 7^9 \cdot 4^{-13} = 7^9 \cdot \frac{1}{4^{13}} = \frac{7^9}{4^{13}} \]
-
Rewrite the entire expression: Substituting this into our expression: \[ \frac{4^{-15} \cdot 7^{10}}{\frac{7^9}{4^{13}}} = 4^{-15} \cdot 7^{10} \cdot \frac{4^{13}}{7^9} \] Which simplifies to: \[ 4^{-15 + 13} \cdot 7^{10 - 9} = 4^{-2} \cdot 7^{1} \]
-
Convert to positive exponents: Rewriting \(4^{-2}\) as \(\frac{1}{4^2}\): \[ 4^{-2} \cdot 7^1 = \frac{7}{4^2} = \frac{7}{16} \]
Thus, the final simplified expression, without any remaining negative exponents and as a fraction, is: \[ \frac{7}{16} \]