Question

simplify.
\\((-2x^{3}yz^{3})(3x^{3}y^{2}z)^{3}\\)

Explanation:

Step1: Simplify the power of the second term

First, we apply the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n=a^{mn}\) to \((3x^{3}y^{2}z)^{3}\).
\[

$$\begin{align*} (3x^{3}y^{2}z)^{3}&=3^{3}(x^{3})^{3}(y^{2})^{3}z^{3}\\ &=27x^{9}y^{6}z^{3} \end{align*}$$

\]

Step2: Multiply the two monomials

Now we multiply \((-2x^{3}yz^{3})\) with \(27x^{9}y^{6}z^{3}\). We use the product rule for exponents \(a^{m}\cdot a^{n}=a^{m + n}\) for the same base.
\[

$$\begin{align*} (-2x^{3}yz^{3})(27x^{9}y^{6}z^{3})&=(-2\times27)(x^{3}\cdot x^{9})(y\cdot y^{6})(z^{3}\cdot z^{3})\\ &=- 54x^{3 + 9}y^{1+6}z^{3 + 3}\\ &=-54x^{12}y^{7}z^{6} \end{align*}$$

\]

Answer:

\(-54x^{12}y^{7}z^{6}\)