Question

\frac{(-4n^{5})^{4}}{-4n^{7}}

ttempt 1 out of 2

Explanation:

Step1: Simplify the numerator using power of a product rule

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

Step2: Simplify the fraction

Now we have the fraction \(\frac{256n^{20}}{-4n^7}\). We can split this into the product of the coefficients and the product of the variables. For the coefficients, \(\frac{256}{-4}=-64\). For the variables, using the quotient rule for exponents \(\frac{a^m}{a^n}=a^{m - n}\), so \(\frac{n^{20}}{n^7}=n^{20 - 7}=n^{13}\). Multiplying these together, we get \(-64n^{13}\).

Answer:

\(-64n^{13}\)