5. Simplify: \( \frac{a^{2} \times c^{3}}{a^{3} \times c^{2}} \frac{c^{3} \times a^{2}}{a^{4} \times c^{2} x d x e} \)

To simplify the expression \( \frac{a^{2} \times c^{3}}{a^{3} \times c^{2}} \frac{c^{3} \times a^{2}}{a^{4} \times c^{2} x d x e} \), let's break it down step by step. First, multiply the two fractions: \[ \frac{a^{2} \cdot c^{3}}{a^{3} \cdot c^{2}} \cdot \frac{c^{3} \cdot a^{2}}{a^{4} \cdot c^{2} \cdot x \cdot d \cdot x \cdot e} \] This yields: \[ \frac{a^{2} \cdot c^{3} \cdot c^{3} \cdot a^{2}}{a^{3} \cdot c^{2} \cdot a^{4} \cdot c^{2} \cdot x \cdot d \cdot x \cdot e} \] Now, combine the numerators and denominators: Numerator: \( a^{2} \cdot a^{2} \cdot c^{3} \cdot c^{3} = a^{4} \cdot c^{6} \) Denominator: \( a^{3} \cdot a^{4} \cdot c^{2} \cdot c^{2} \cdot x \cdot d \cdot x \cdot e = a^{7} \cdot c^{4} \cdot x^{2} \cdot d \cdot e \) Now we have: \[ \frac{a^{4} \cdot c^{6}}{a^{7} \cdot c^{4} \cdot x^{2} \cdot d \cdot e} \] Next, apply the properties of exponents: \[ = \frac{1}{a^{7-4} \cdot c^{4-6} \cdot x^{2} \cdot d \cdot e} = \frac{1}{a^{3} \cdot c^{-2} \cdot x^{2} \cdot d \cdot e} \] This simplifies further as \( c^{-2} = \frac{1}{c^{2}} \), so: \[ = \frac{c^{2}}{a^{3} \cdot x^{2} \cdot d \cdot e} \] Thus, the final simplified result is: \[ \frac{c^{2}}{a^{3} \cdot x^{2} \cdot d \cdot e} \]