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:
\(5^{-4}(w^{-4})^{-4}(h^{5})^{-4}\)

Step2: Simplify the exponents using the power of a power rule

The power of a power rule is \((a^m)^n = a^{mn}\). So:
\(5^{-4}w^{(-4)\times(-4)}h^{5\times(-4)} = 5^{-4}w^{16}h^{-20}\)
Now the numerator is \(5^{-4}w^{16}h^{-20}\) and the denominator is \(4w^{2}\), so the expression becomes \(\frac{5^{-4}w^{16}h^{-20}}{4w^{2}}\)

Step3: Apply the quotient rule for exponents

The quotient rule for exponents is \(\frac{a^m}{a^n}=a^{m - n}\). For the \(w\) terms: \(w^{16-2}=w^{14}\). For the \(h\) term, \(h^{-20}\) can be moved to the denominator as \(h^{20}\) (since \(a^{-n}=\frac{1}{a^{n}}\)). Also, \(5^{-4}=\frac{1}{5^{4}}=\frac{1}{625}\)
So now we have \(\frac{w^{14}}{4\times5^{4}w^{2}h^{20}}\)? Wait, no, let's re - do this step.
The original expression after step 2 is \(\frac{5^{-4}w^{16}h^{-20}}{4w^{2}}\). We can rewrite this as \(\frac{5^{-4}}{4}\times\frac{w^{16}}{w^{2}}\times h^{-20}\)
Using the quotient rule for \(w\): \(\frac{w^{16}}{w^{2}}=w^{16 - 2}=w^{14}\)
And \(h^{-20}=\frac{1}{h^{20}}\), \(5^{-4}=\frac{1}{5^{4}}=\frac{1}{625}\)
So putting it all together: \(\frac{w^{14}}{4\times625h^{20}}\) (Wait, no, let's correct the coefficient part. \(5^{-4}=\frac{1}{5^{4}}\), so \(\frac{5^{-4}}{4}=\frac{1}{4\times5^{4}}=\frac{1}{4\times625}=\frac{1}{2500}\))
Wait, actually, let's start over with the coefficient and each variable:
\(\frac{(5w^{-4}h^{5})^{-4}}{4w^{2}}=\frac{5^{-4}w^{16}h^{-20}}{4w^{2}}\)
Now, using the rule \(a^{-n}=\frac{1}{a^{n}}\) for \(h^{-20}\), we get \(\frac{5^{-4}w^{16}}{4w^{2}h^{20}}\)
Then, using \(\frac{a^{m}}{a^{n}}=a^{m - n}\) for \(w\): \(\frac{w^{16}}{w^{2}}=w^{16-2}=w^{14}\)
And \(5^{-4}=\frac{1}{5^{4}}=\frac{1}{625}\), so substituting back, we have \(\frac{w^{14}}{4\times625h^{20}}=\frac{w^{14}}{2500h^{20}}\)

Answer:

\(\frac{w^{16}}{2500w^{2}h^{20}}\) (Wait, no, let's re - calculate the coefficient correctly. \(5^{-4}=\frac{1}{5^{4}}=\frac{1}{625}\), and the denominator has a 4, so \(\frac{1}{625\times4}=\frac{1}{2500}\), and \(w^{16}\div w^{2}=w^{14}\), \(h^{-20}=\frac{1}{h^{20}}\), so the correct answer is \(\frac{w^{14}}{2500h^{20}}\))