a. \( \left[\frac{(-2)^{5} \cdot(-2)^{8}}{(-2)^{15}}\right]^{-4} \) b. \( \left[(-3)^{4}\right]^{3} \cdot(-2)^{5} \cdot(-3)^{12} \cdot(-2)^{2} \) c. \( \left[\frac{(-7)^{4} \cdot(3)^{10} \cdot(-7)^{6}}{(-3)^{11} \cdot(7)^{11}}\right]^{5} \div(3)^{-2} \)
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Let’s break these expressions down and see what fun facts we can learn! Using exponent rules, the first expression simplifies nicely. Combining powers, we have \( \left[(-2)^{5 + 8 - 15}\right]^{-4} = \left[(-2)^{-2}\right]^{-4} \), which becomes \( (-2)^{8} = 256 \). Negative numbers raised to even powers yield positive results, keeping our math on the bright side! For the second expression, notice how \( \left[(-3)^{4}\right]^{3} \) means we’ll have \( (-3)^{12} \). Combine like bases and gather all the terms: \( (-3)^{12} \cdot (-3)^{12} \cdot (-2)^{7} = (-3)^{24} \cdot (-2)^{7} \), but remember that \( (-3)^{24} \) is a big positive number since it's an even power! So multiply away! Both of these show the power of exponents, not just in simplifying expressions but also in showing the foundational rules of mathematics in action. If you mix up positive and negative numbers, staying mindful of those even and odd powers will steer you clear of pesky mistakes!