5) \( \left(1,000^{\frac{3}{2}}\right)^{-\frac{4}{9}} \)

To tackle the expression \( \left(1,000^{\frac{3}{2}}\right)^{-\frac{4}{9}} \), you can simplify it step-by-step. First, rewrite \( 1,000 \) as \( 10^3 \), so the expression becomes \( \left((10^3)^{\frac{3}{2}}\right)^{-\frac{4}{9}} \). Apply the power of a power rule which states \( (a^m)^n = a^{m \cdot n} \). This gives you \( (10^{3 \cdot \frac{3}{2}})^{-\frac{4}{9}} = (10^{\frac{9}{2}})^{-\frac{4}{9}} = 10^{\frac{9}{2} \cdot -\frac{4}{9}} \). Calculate the exponent \( \frac{9}{2} \cdot -\frac{4}{9} = -2 \) leading to \( 10^{-2} \). Therefore, the final answer is \( \frac{1}{100} \) or \( 0.01 \). In mathematics, understanding exponents can help simplify complex calculations and equations, making problem-solving much swifter! Engaging with exponents in real-life situations, like measuring interest over time or calculating areas, can create practical examples that enliven these concepts beyond the classroom. Don't shy away from mixing numbers with exponents in everyday life!