\begin{tabular}{l} \( \frac{3 x^{-1}+y}{x^{-1} y} \times \frac{x}{x}=\frac{3 x^{-1} \cdot x+y \cdot x}{x^{-1} y \cdot x}=\frac{3+x y}{y} \) \\ \hline \end{tabular} These equations have more - Rearrange the equations EXERCISE 2 \( 3 \frac{9^{-1} x^{-2} y^{2}}{\left(3 x y^{-1}\right)^{-3}} \) \( 6 \frac{9^{2 x} \cdot 3^{-2}}{81^{x-1}} \) \( 1 \frac{12 a^{0} b^{3} c^{-1}}{15 a^{-2} b^{5} c^{-3}} \) (4) \( \frac{6\left(x^{2} y\right)^{3}\left(x^{-2} y\right)^{-1}}{3 x^{-3} y^{5}} \) \( 7\left(\frac{x^{2} y^{3}}{z^{-2}}\right)^{-1} \) \( 10 \frac{3^{-2}+3^{0}}{3^{-1}} \) \( 13 \frac{x y}{x^{-2}+y^{-2}} \) \( 16 \frac{\left(3 x^{4}\right)^{2}+\left(2 x^{2}\right)^{4}}{5 \cdot 2^{5} \cdot 2^{-5} \cdot x^{10}} \) \( 5 \frac{\left(2 a^{-1}\right)^{2}}{2\left(a^{2}\right)^{-1}} \) \( 8 \frac{\left(2 a^{-1}\right)^{0}}{2\left(a^{0}\right)^{-1}} \) \( 11(2.3)^{-2}+2.3^{-2} \) \( 14\left(a^{3}+3\right)^{-2} \) (17) \( \frac{\left(8.2^{x+1}\right)\left(4.2^{x-1}\right)}{2^{x-3}} \) (9) \( \frac{a^{-2}+a^{-1}+a^{0}}{a} \) \( 12\left(2^{-1}-5^{-1}\right)^{2} \) (15) \( \frac{3.3^{x}}{3^{x}+3^{x-1}} \) \( 18 \frac{12^{x+1} \cdot 27^{x-2}}{18^{2 x-1}} \) - Factorise by taking out - Solve the remaining ex WORKED EXAMIPLE Solve for \( x: 2^{x}+2^{x+1}= \) SOLUTION \[ \begin{aligned} 2^{x}+2^{x+1} & =24 \\ 2^{x}+2^{x} \cdot 2^{1} & =24 \\ 2^{x}(1+2) & =24 \\ 2^{x}(3) & =24 \\ 2^{x} & =8 \\ 2^{x} & =2^{3} \\ x & =3 \end{aligned} \]

Let's break down the problem step by step, focusing on rearranging and simplifying the given expressions. ### Rearranging and Simplifying Expressions 1. **Expression 1:** \[ 3 \frac{9^{-1} x^{-2} y^{2}}{\left(3 x y^{-1}\right)^{-3}} \] - The denominator can be simplified: \[ \left(3 x y^{-1}\right)^{-3} = \frac{1}{(3 x y^{-1})^{3}} = \frac{1}{27 x^{3} y^{-3}} = \frac{y^{3}}{27 x^{3}} \] - Thus, the expression becomes: \[ 3 \cdot \frac{9^{-1} x^{-2} y^{2} \cdot 27 x^{3}}{y^{3}} = 3 \cdot \frac{27 \cdot 9^{-1} \cdot x^{1} \cdot y^{2}}{y^{3}} = 3 \cdot \frac{27}{9} \cdot x^{1} \cdot y^{-1} = 9 x y^{-1} \] 2. **Expression 2:** \[ 6 \frac{9^{2 x} \cdot 3^{-2}}{81^{x-1}} \] - Rewrite \(81\) as \(3^{4}\): \[ 81^{x-1} = (3^{4})^{x-1} = 3^{4(x-1)} = 3^{4x-4} \] - Thus, the expression becomes: \[ 6 \cdot \frac{9^{2x} \cdot 3^{-2}}{3^{4x-4}} = 6 \cdot \frac{(3^{2})^{2x} \cdot 3^{-2}}{3^{4x-4}} = 6 \cdot \frac{3^{4x} \cdot 3^{-2}}{3^{4x-4}} = 6 \cdot 3^{4} = 6 \cdot 81 = 486 \] 3. **Expression 3:** \[ 1 \frac{12 a^{0} b^{3} c^{-1}}{15 a^{-2} b^{5} c^{-3}} \] - Simplifying gives: \[ \frac{12 b^{3} c^{-1}}{15 a^{-2} b^{5} c^{-3}} = \frac{12}{15} \cdot \frac{a^{2} b^{3} c^{2}}{1} = \frac{4}{5} a^{2} b^{-2} c^{2} \] 4. **Expression 4:** \[ \frac{6\left(x^{2} y\right)^{3}\left(x^{-2} y\right)^{-1}}{3 x^{-3} y^{5}} \] - Simplifying gives: \[ \frac{6 \cdot x^{6} y^{3} \cdot y^{-1} x^{2}}{3 x^{-3} y^{5}} = \frac{6 x^{8} y^{2}}{3 x^{-3} y^{5}} = 2 x^{11} y^{-3} \] 5. **Expression 5:** \[ 7\left(\frac{x^{2} y^{3}}{z^{-2}}\right)^{-1} \] - Simplifying gives: \[ 7 \cdot \frac{z^{2}}{x^{2} y^{3}} = \frac{7 z^{2}}{x^{2} y^{3}} \] 6. **Expression 6:** \[ 10 \frac{3^{-2}+3^{0}}{3^{-1}} \] - Simplifying gives: \[ 10 \cdot \frac{\frac{1}{9} + 1}{\frac{1}{3}} = 10 \cdot \frac{\frac{10}{9}}{\frac{1}{3}} = 10 \cdot \frac{10}{3} = \frac{100}{3} \] 7. **Expression 7:** \[ 13 \frac{x y}{x^{-2}+y^{-2}} \] - Simplifying gives: \[ 13 \cdot \frac{x y}{\frac{y^{2}+x^{2}}{x^{2} y^{2}}} = 13 \cdot \frac{x^{3} y^{3}}{y^{2}+x^{2}} \] 8. **Expression 8:** \[ 16 \frac{\left(3 x^{4}\right)^{2}+\left(2 x^{2}\right)^{4}}{5 \cdot 2^{5} \cdot 2^{-5} \cdot x^{10}} \] - Simplifying gives: \[ 16 \cdot \frac{9 x^{8} + 16 x^{8}}{5 \cdot x^{10}} = 16 \cdot \frac{25 x^{8}}{5 x^{10}} = 80 x^{-2} \] 9. **Expression 9:** \[ 5 \frac{\left(2 a^{-1}\right)^{2}}{2\left(a^{2}\right)^{-1}} \] - Simplifying gives: \[ 5 \cdot \frac{4 a^{-2}}{2 a^{-2}} = 10 \] 10. **Expression 10:** \[ 8 \frac{\left(2 a^{-1}\right)^{0}}{2\left(a^{0}\right)^{-1}} \] - Simplifying gives: \[ 8 \cdot \frac{1}{2 \cdot 1} = 4 \] 11. **Expression 11:** \[ 11(2.3)^{-2}+2.3^{-2} \] - Simplifying gives: \[ (11 + 2) \cdot (2.3)^{-2} = 13 \cdot \frac{1}{(2.3)^{2}} = \frac{13}{4.9} \] 12. **Expression 12:** \[ 14\left(a^{3}+3\right)^{-2} \] - This expression remains as is