Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{(5w^{-4}h^{5})^{-4}}{4w^{2}}\\)

Explanation:

Step1: Apply the power of a product rule to the numerator

The power of a product rule states that \((ab)^n = a^n b^n\). For \((5w^{-4}h^{5})^{-4}\), we have:
\(5^{-4}(w^{-4})^{-4}(h^{5})^{-4}\)
Using the power of a power rule \((a^m)^n=a^{mn}\), we get:
\(5^{-4}w^{(-4)\times(-4)}h^{5\times(-4)} = 5^{-4}w^{16}h^{-20}\)
So the numerator becomes \(5^{-4}w^{16}h^{-20}\) and the expression is now \(\frac{5^{-4}w^{16}h^{-20}}{4w^{2}}\)

Step2: Apply the quotient rule for exponents

The quotient rule for exponents is \(\frac{a^m}{a^n}=a^{m - n}\). For the \(w\) terms: \(\frac{w^{16}}{w^{2}}=w^{16 - 2}=w^{14}\)
For the \(h\) term, we know that \(a^{-n}=\frac{1}{a^{n}}\), so \(h^{-20}=\frac{1}{h^{20}}\)
For the constant term \(5^{-4}\), we know that \(a^{-n}=\frac{1}{a^{n}}\), so \(5^{-4}=\frac{1}{5^{4}}=\frac{1}{625}\)
Now the expression is \(\frac{1}{625}\times\frac{w^{14}}{4}\times\frac{1}{h^{20}}\)

Step3: Combine the constants and simplify

Multiply the constants \(\frac{1}{625}\) and \(\frac{1}{4}\) to get \(\frac{1}{2500}\)
Putting it all together, we have \(\frac{w^{14}}{2500h^{20}}\)

Answer:

\(\frac{w^{14}}{2500h^{20}}\)