(4) \( \quad-3\left(-2 a^{3}\right)^{2}+2\left(-2 a^{2}\right)^{3} \)

Let’s simplify the expression step by step. First, we start with the first term: \[ -3\left(-2 a^{3}\right)^{2} = -3 \cdot 4a^{6} = -12a^{6} \] Now, for the second term: \[ 2\left(-2 a^{2}\right)^{3} = 2 \cdot (-8 a^{6}) = -16 a^{6} \] Combining both results: \[ -12a^{6} - 16a^{6} = -28a^{6} \] Thus, the simplified expression is: \[ -28a^{6} \] Now, let’s add some context! Did you know that the power rules for exponents are one of the cornerstones of algebra? These rules allow us to manipulate and simplify expressions elegantly, making problem-solving quicker and more efficient. It’s like learning a secret language that mathematicians and scientists around the world use to communicate complex ideas succinctly! When it comes to applying exponent rules, a common mistake is to forget to apply the negative sign correctly. For instance, when cubing \(-2 a^2\), it’s crucial to remember that both the coefficient and the variable are affected by the exponent, leading sometimes to errors in signs. So, always double-check your work to avoid a pesky negative slipping through the cracks!