Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{(4w^{-1}c^{4})^{-5}}{5c^{-3}}\\)

Explanation:

Step1: Apply power of a product rule

For the numerator \((4w^{-1}c^{4})^{-5}\), use \((ab)^n = a^n b^n\) and \((a^m)^n=a^{mn}\). So we get \(4^{-5}(w^{-1})^{-5}(c^{4})^{-5}=4^{-5}w^{5}c^{-20}\).

Step2: Rewrite the fraction

The expression becomes \(\frac{4^{-5}w^{5}c^{-20}}{5c^{-3}}\). Now, use the rule \(\frac{a^m}{a^n}=a^{m - n}\) for the \(c\) terms and \(\frac{1}{a^n}=a^{-n}\) for the \(4^{-5}\) term.
First, handle the \(c\) terms: \(c^{-20-(-3)} = c^{-17}\). Then, \(4^{-5}=\frac{1}{4^{5}}=\frac{1}{1024}\). So the expression is \(\frac{w^{5}c^{-17}}{5\times4^{5}}\).

Step3: Convert negative exponents to positive

Using \(a^{-n}=\frac{1}{a^n}\), we rewrite \(c^{-17}\) as \(\frac{1}{c^{17}}\) and \(4^{-5}\) is already handled. So we have \(\frac{w^{5}}{5\times4^{5}\times c^{17}}\). Calculate \(4^{5}=1024\), so \(5\times4^{5} = 5120\).

Answer:

\(\frac{w^{5}}{5120c^{17}}\)