\( 3.4 \quad \frac { - 21 x ^ { 2 } y ^ { - 3 } z ^ { 5 } } { 3 x ^ { 3 } y ^ { 4 } z ^ { 2 } } \)

To simplify the expression \( \frac{-21 x^{2} y^{-3} z^{5}}{3 x^{3} y^{4} z^{2}} \), we can start by dividing the coefficients and then simplifying the variable parts separately. The coefficient \(-21\) divided by \(3\) simplifies to \(-7\). Next, for the variables: - For \(x^{2}\) divided by \(x^{3}\), we subtract the exponents: \(x^{2 - 3} = x^{-1} = \frac{1}{x}\). - For \(y^{-3}\) divided by \(y^{4}\), we again subtract the exponents: \(y^{-3 - 4} = y^{-7} = \frac{1}{y^{7}}\). - For \(z^{5}\) divided by \(z^{2}\), we subtract the exponents as well: \(z^{5 - 2} = z^{3}\). Putting it all together, we have: \[ -7 \cdot \frac{1}{x} \cdot \frac{1}{y^{7}} \cdot z^{3} = \frac{-7 z^{3}}{x y^{7}}. \] So the simplified form is: \[ \frac{-7 z^{3}}{x y^{7}}. \]