To simplify the expression \((4^3 \cdot 7^{-2})^{-5} / (7^9 \cdot 4^{-13})\), we will break it down step by step:
-
Start with simplifying the numerator: \[ (4^3 \cdot 7^{-2})^{-5} = (4^3)^{-5} \cdot (7^{-2})^{-5} \] Now apply the power of a power rule \((a^m)^n = a^{m \cdot n}\): \[ (4^3)^{-5} = 4^{-15} \quad \text{and} \quad (7^{-2})^{-5} = 7^{10} \] So we have: \[ (4^3 \cdot 7^{-2})^{-5} = 4^{-15} \cdot 7^{10} \]
-
Now substituting back into the original expression gives: \[ \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} \]
-
In the fraction, simplify by subtracting the exponents of like bases:
-
For base \(4\): \[ 4^{-15} / 4^{-13} = 4^{-15 - (-13)} = 4^{-15 + 13} = 4^{-2} \]
-
For base \(7\): \[ 7^{10} / 7^9 = 7^{10 - 9} = 7^1 = 7 \]
-
-
Putting it all together: \[ \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} = 4^{-2} \cdot 7 \]
-
Rewrite \(4^{-2}\) as \(1/4^2\): \[ 4^{-2} \cdot 7 = \frac{7}{4^2} = \frac{7}{16} \]
Thus, the simplified form of the expression is: \[ \frac{7}{16} \]