How to find $\arg(z)$ and $|z|$? How to find $|z|$ and $\arg (z)$
$z$ is complex number and
$z$ is defined by $$z=\left(\cos\frac\pi5+i\sin\frac\pi5\right)^{15}\cdot(3-3i)^{20}$$
I`ve tried to behave it like $$e^{i15\frac\pi5}\cdot e^{i20\frac\pi4}$$ and got in result = $$e^{i3\pi}\cdot e^{i5\pi}=e^{i8\pi}$$ which gives me $$(-1)^8=1$$ and if $$+z=|z|$$ so $$|z|=1$$ So is it my answer and how to find? $$Arg(z)$$

Just remember some basic properies of $\arg$ and absolue value: If a value is given in polar form, then the polar coordinates are obtained.  With $r, \varphi\in \Bbb R$ and $r\geqslant 0$:
$$\begin{align*}
|r\cdot e^{i\varphi}| &= r \\
\arg (r\cdot e^{i\varphi}) &\equiv \varphi 
\end{align*}$$
Where $\equiv$ means that the value is only determined up to an integer multiple of $2\pi$.  Common choices for $\arg$ are $0\leqslant\arg<2\pi$  or $-\pi < \arg\leqslant \pi$.
Moreover, for a product of two complex numbers we have:
$$\begin{align}
|z\cdot w| &= |z|\cdot|w| \\
\arg(z\cdot w) &\equiv \arg z + \arg w \\
\end{align}$$
which implies for $z\neq 0$ and real exponents $p\neq 0$
$$\begin{align}
|z^p| &= |z|^p \\
\arg(z^p) &\equiv p\cdot\arg z\\
\end{align}$$
So the first lines for your $\arg z$ would prepare for easy computation and read:
$$\begin{align}
z &= \left(\cos\frac\pi5+i\sin\frac\pi5\right)^{15}\!\!\cdot\,(3-3i)^{20} \\
&= \left(\exp\frac{\pi i}5\right)^{15}\!\!\cdot\,3^{20}\cdot(1-i)^{20} \\
\end{align}$$
Hence:
$$\begin{align}
\arg z &\equiv 15\cdot \frac\pi5 + 20\cdot\underbrace{\arg(3)}_{\textstyle=0} + 20\cdot\underbrace{\arg(1-i)}_{\textstyle=-\pi/4} \\
&\equiv 3\pi - 5\pi \equiv 0 \mod 2\pi
\end{align}$$
