Question

write a simplified expression that is equivalent to ((4x^{2}y^{5})^{2}(3x^{4}y^{2})^{4})

Explanation:

Step1: Apply power of a product rule

For \((4x^{2}y^{5})^{2}\), we use \((ab)^n=a^nb^n\) and \((a^m)^n = a^{mn}\). So \((4x^{2}y^{5})^{2}=4^{2}(x^{2})^{2}(y^{5})^{2}=16x^{4}y^{10}\).
For \((3x^{4}y^{2})^{4}\), we have \(3^{4}(x^{4})^{4}(y^{2})^{4}=81x^{16}y^{8}\).

Step2: Multiply the two simplified expressions

Now we multiply \(16x^{4}y^{10}\) and \(81x^{16}y^{8}\). Using the rule \(a^m\times a^n=a^{m + n}\) for the same base, we get:
For the coefficients: \(16\times81 = 1296\).
For the \(x\) terms: \(x^{4}\times x^{16}=x^{4 + 16}=x^{20}\).
For the \(y\) terms: \(y^{10}\times y^{8}=y^{10 + 8}=y^{18}\).
So the product is \(1296x^{20}y^{18}\).

Answer:

\(1296x^{20}y^{18}\)