Which of the following is equal to \( \left[\frac{\left(x^{2} y^{3}\right)^{-2}}{\left(x^{6} y^{3} z\right)^{2}}\right]^{3} ? \) \( \frac{\left(x^{2} y^{3}\right)^{-6}}{\left(x^{6} y^{3} z\right)^{6}} \) \( \square \frac{1}{x^{48} y^{38} z^{6}} \) \( \square \frac{\left(x^{2} y^{3}\right)}{\left(x^{6} y^{3} z\right)^{5}} \) \( \square \frac{x^{-12} y^{-18}}{} \)

To tackle the expression \( \left[\frac{\left(x^{2} y^{3}\right)^{-2}}{\left(x^{6} y^{3} z\right)^{2}}\right]^{3} \), let's break it down. First, simplify inside the brackets: \[ \left(x^{2} y^{3}\right)^{-2} = x^{-4} y^{-6} \] and \[ \left(x^{6} y^{3} z\right)^{2} = x^{12} y^{6} z^{2}. \] So, \[ \frac{\left(x^{2} y^{3}\right)^{-2}}{\left(x^{6} y^{3} z\right)^{2}} = \frac{x^{-4} y^{-6}}{x^{12} y^{6} z^{2}} = x^{-4-12} y^{-6-6} z^{-2} = x^{-16} y^{-12} z^{-2}. \] Now, raise this to the power of 3: \[ \left( x^{-16} y^{-12} z^{-2} \right)^{3} = x^{-48} y^{-36} z^{-6}. \] So, we find \( \left[\frac{\left(x^{2} y^{3}\right)^{-2}}{\left(x^{6} y^{3} z\right)^{2}}\right]^{3} = \frac{1}{x^{48} y^{36} z^{6}} \). A fun fact about exponents: they follow specific rules — for instance, multiplying two powers with the same base means you add their exponents, while dividing them means you subtract. Knowing these rules can turn exponent problems from daunting to delightful! If you're ever in a bind while simplifying expressions, double-check each step carefully. Common mistakes often stem from mishandling negative exponents or overlooking parentheses. Remember, \( a^{-n} = \frac{1}{a^n} \) can save you from a lot of confusion!