Question

simplify.
$(-4x^4y^5)^5$

Explanation:

Step1: Apply power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So we can apply this to \((-4x^{4}y^{5})^{5}\) as follows:
\((-4x^{4}y^{5})^{5}=(-4)^{5}(x^{4})^{5}(y^{5})^{5}\)

Step2: Calculate \((-4)^5\)

We know that \((-4)^5 = (-4)\times(-4)\times(-4)\times(-4)\times(-4)\). Since there are 5 (an odd number) negative factors, the result will be negative. And \(4^5 = 1024\), so \((-4)^5=-1024\).

Step3: Apply power of a power rule to \((x^{4})^{5}\) and \((y^{5})^{5}\)

The power of a power rule is \((a^m)^n=a^{m\times n}\). For \((x^{4})^{5}\), we have \(x^{4\times5}=x^{20}\). For \((y^{5})^{5}\), we have \(y^{5\times5}=y^{25}\).

Step4: Combine the results

Putting it all together, we have \((-4)^{5}(x^{4})^{5}(y^{5})^{5}=-1024x^{20}y^{25}\)

Answer:

\(-1024x^{20}y^{25}\)