To simplify \((4x^5 y^2 z^7)^2\), we apply the power of a product rule, which states that \((a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n\).
Applying that rule here, we get:
\[ (4x^5 y^2 z^7)^2 = 4^2 (x^5)^2 (y^2)^2 (z^7)^2 \]
Calculating each part:
- \(4^2 = 16\)
- \((x^5)^2 = x^{5 \cdot 2} = x^{10}\)
- \((y^2)^2 = y^{2 \cdot 2} = y^4\)
- \((z^7)^2 = z^{7 \cdot 2} = z^{14}\)
Putting it all together, we have:
\[ (4x^5 y^2 z^7)^2 = 16x^{10}y^4z^{14} \]
Thus, the simplified expression is:
\[ \boxed{16x^{10}y^4z^{14}} \]