Simplify: (a^b * b^a)/(c^b) * b^-c * 1/a Fraction Expression

To solve the given problem, we'll apply the laws of exponents to simplify the expression $\frac{a^b b^a}{c^b} \cdot b^{-c} \cdot \frac{1}{a}$.

Let's go through each step:

• Start with the expression $\frac{a^b b^a}{c^b} \cdot b^{-c} \cdot \frac{1}{a}$.
• Rewrite $b^{-c}$ as $\frac{1}{b^c}$ using the rule $x^{-n} = \frac{1}{x^n}$.
• Substitute it back: $\frac{a^b b^a}{c^b} \cdot \frac{1}{b^c} \cdot \frac{1}{a}$.
• Combine the expressions into a single fraction: $\frac{a^b b^a}{c^b \cdot b^c \cdot a}$.
• Use the rule $x^m \cdot x^n = x^{m+n}$ to simplify the exponents in the numerator and denominator: - In the numerator, no changes necessary since terms are already separated. - In the denominator, combine $b^a$ and $b^c$ using the exponent rule: $b^{a+c}$.
• Update the fraction: $\frac{a^b}{a} \cdot \frac{b^a}{b^{a+c}} \cdot \frac{1}{c^b}$.
• Simplify each component: - $\frac{a^b}{a} = a^{b-1}$, - $\frac{b^a}{b^{a+c}} = b^{a-(a+c)} = b^{-c} = \frac{1}{b^c}$.
• Combine all components: $\frac{a^{b-1}}{c^b \cdot b^c}$.
• Express the result combining all simplified terms: $\frac{1}{a^{1-b} b^{c-a} c^b}$.

Therefore, the solution to the problem is $\frac{1}{a^{1-b} b^{c-a} c^b}$.