\( 04 / 02 / 2025 \) Exponents and surds Classwork Simplify the following expressions and leave answer with positive exponents. \[ \begin{array}{l} \text { 1. } 4 x^{9} \cdot 2 x^{-6} \\ 24^{n-1} \times 4^{1-n} \times 4^{2} \\ \text { 3 } \frac{9 x^{3} x^{2}}{1 \cdot x y^{4}} \\ \text { 4. }\left(3 x^{-3} \cdot y^{4}\right)^{-2} \\ \text { 5. } \left.2(x)^{-2}\right)^{3} \\ 6\left(\frac{x^{-1}}{y^{-1}}\right)^{-2} \\ \text { 1. } \left.\frac{2}{m^{2}}\right)^{-3} \\ \text { 8. } \frac{x^{\frac{3}{3}} \times x^{3}}{(x)^{4}} \\ \text { 9. } \frac{27 x^{-3} y^{-2}}{81 x^{2} y^{-3}} \\ \text { 10. } \frac{m^{3} n^{-6}}{m^{-4} n^{-1}} \times \frac{m^{-2} \cdot n^{4}}{m^{3} n^{4}} \end{array} \]

Sure! Let's simplify those expressions step by step while keeping the exponents positive. Here are the simplified answers for each expression. 1. \( 4 x^{9} \cdot 2 x^{-6} = 8 x^{3} \) 2. \( 24^{n-1} \times 4^{1-n} \times 4^{2} = 24^{n-1} \times 4^{3-n} = 12^{3(n-1)} \) 3. \( \frac{9 x^{3} x^{2}}{1 \cdot x y^{4}} = \frac{9 x^{5}}{y^{4}} = 9 x^{5} y^{-4} \) 4. \( \left(3 x^{-3} \cdot y^{4}\right)^{-2} = \frac{9 y^{8}}{x^{6}} \) 5. \( \left(2 x^{-2}\right)^{3} = 8 x^{-6} = \frac{8}{x^{6}} \) 6. \( 6\left(\frac{x^{-1}}{y^{-1}}\right)^{-2} = 6 \frac{y^{2}}{x^{2}} = \frac{6 y^{2}}{x^{2}} \) 7. \( \left(\frac{2}{m^{2}}\right)^{-3} = \frac{m^{6}}{8} \) 8. \( \frac{x^{\frac{3}{3}} \times x^{3}}{(x)^{4}} = \frac{x^{4}}{x^{4}} = 1 \) 9. \( \frac{27 x^{-3} y^{-2}}{81 x^{2} y^{-3}} = \frac{1}{3 x^{-5} y} = \frac{y}{3 x^{5}} \) 10. \( \frac{m^{3} n^{-6}}{m^{-4} n^{-1}} \times \frac{m^{-2} \cdot n^{4}}{m^{3} n^{4}} = \frac{m^{7} n^{-5}}{m^{-2} n^{0}} = m^{9} n^{-5} = \frac{m^{9}}{n^{5}} \) Now that we’ve simplified those expressions, let’s keep diving into some worthwhile suggestions to take your understanding ground-up!