Question

1. write a simplified expression that is equivalent to \\(\frac{9a^{-5}b^{4}c^{8}}{27a^{-7}b^{-9}c^{12}}\\) for all values of \\(a\\), \\(b\\), and \\(c\\) where the expression is defined.

Explanation:

Step1: Simplificar coeficientes numéricos

$\frac{9}{27} = \frac{1}{3}$

Step2: Simplificar términos con $a$

$\frac{a^{-5}}{a^{-7}} = a^{-5 - (-7)} = a^{2}$

Step3: Simplificar términos con $b$

$\frac{b^{4}}{b^{-9}} = b^{4 - (-9)} = b^{13}$

Step4: Simplificar términos con $c$

$\frac{c^{8}}{c^{12}} = c^{8 - 12} = c^{-4} = \frac{1}{c^{4}}$

Step5: Combinar todos los términos

$\frac{1}{3} \cdot a^{2} \cdot b^{13} \cdot \frac{1}{c^{4}}$

Answer:

$\frac{a^{2}b^{13}}{3c^{4}}$