Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{(-4n^{5})^{4}}{-4n^{7}}\\)

Explanation:

Step1: Simplify the numerator using power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So for \((-4n^5)^4\), we have \((-4)^4\times(n^5)^4\).
\((-4)^4 = 256\) and using the power of a power rule \((a^m)^n=a^{mn}\), \((n^5)^4 = n^{5\times4}=n^{20}\). So the numerator becomes \(256n^{20}\).

Step2: Simplify the fraction by dividing coefficients and using exponent rule for division

The fraction is \(\frac{256n^{20}}{-4n^7}\). First, divide the coefficients: \(\frac{256}{-4}=-64\). Then, for the variables, use the rule \(\frac{a^m}{a^n}=a^{m - n}\), so \(\frac{n^{20}}{n^7}=n^{20 - 7}=n^{13}\).

Step3: Combine the results

Multiply the coefficient and the variable part together: \(-64\times n^{13}=-64n^{13}\)

Answer:

\(-64n^{13}\)