3.1.5. \( -\left(-3 x^{4} y\right)^{3} \)

When you have an expression like \( -\left(-3 x^{4} y\right)^{3} \), you're dealing with both negative signs and exponents. Notably, raising a negative number to an odd power results in a negative number, and here’s where it gets interesting! So, \( \left(-3 x^{4} y\right)^{3} \) gives you \(-27 x^{12} y^{3}\) after applying the power rule to each component. If you then include the leading negative sign from the original expression, you’ll end up with \( 27 x^{12} y^{3} \). It's essential to remember the order of operations when simplifying such expressions. A common mistake here is to neglect the leading negative sign when raising the entire term to a power. Make sure to carry that negative sign through to your final answer! Always double-check to avoid turning a positive into a negative unwittingly.