QUESTION 1 1.1 Fully simplify the following expressions: 1.1.1 \( \frac{19.3^{x}-9.3^{x+2}}{18.3^{x-3}+39.3^{1+x}} \) \( 1.1 .2 \frac{\sqrt{48 x^{6}}-\sqrt[3]{125 x^{9}} \cdot 3^{\frac{1}{2}}}{\sqrt{75 x^{6}}}=\frac{124}{2} \times ? \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\left(\sqrt{48x^{6}}-\left(125x^{9}\right)^{\frac{1}{3}}\times 3^{\frac{1}{2}}\right)}{\sqrt{75x^{6}}}\) - step1: Remove the parentheses: \(\frac{\sqrt{48x^{6}}-\left(125x^{9}\right)^{\frac{1}{3}}\times 3^{\frac{1}{2}}}{\sqrt{75x^{6}}}\) - step2: Simplify the root: \(\frac{4\sqrt{3}\times x^{3}-\left(125x^{9}\right)^{\frac{1}{3}}\times 3^{\frac{1}{2}}}{\sqrt{75x^{6}}}\) - step3: Multiply the terms: \(\frac{4\sqrt{3}\times x^{3}-5\times 3^{\frac{1}{2}}x^{3}}{\sqrt{75x^{6}}}\) - step4: Simplify the root: \(\frac{4\sqrt{3}\times x^{3}-5\times 3^{\frac{1}{2}}x^{3}}{5\sqrt{3}\times x^{3}}\) - step5: Subtract the terms: \(\frac{-\sqrt{3}\times x^{3}}{5\sqrt{3}\times x^{3}}\) - step6: Reduce the fraction: \(\frac{-\sqrt{3}}{5\sqrt{3}}\) - step7: Reduce the fraction: \(\frac{-1}{5}\) - step8: Rewrite the fraction: \(-\frac{1}{5}\) Calculate or simplify the expression \( (19.3^x - 9.3^(x+2)) / (18.3^(x-3) + 39.3^(1+x)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(19.3^{x}-9.3^{x+2}\right)}{\left(18.3^{x-3}+39.3^{1+x}\right)}\) - step1: Remove the parentheses: \(\frac{19.3^{x}-9.3^{x+2}}{18.3^{x-3}+39.3^{1+x}}\) - step2: Convert the expressions: \(\frac{\left(\frac{193}{10}\right)^{x}-9.3^{x+2}}{18.3^{x-3}+39.3^{1+x}}\) - step3: Convert the expressions: \(\frac{\left(\frac{193}{10}\right)^{x}-\left(\frac{93}{10}\right)^{x+2}}{18.3^{x-3}+39.3^{1+x}}\) - step4: Convert the expressions: \(\frac{\left(\frac{193}{10}\right)^{x}-\left(\frac{93}{10}\right)^{x+2}}{\left(\frac{183}{10}\right)^{x-3}+39.3^{1+x}}\) - step5: Convert the expressions: \(\frac{\left(\frac{193}{10}\right)^{x}-\left(\frac{93}{10}\right)^{x+2}}{\left(\frac{183}{10}\right)^{x-3}+\left(\frac{393}{10}\right)^{1+x}}\) - step6: Rewrite the expression: \(\frac{\left(\frac{193}{10}\right)^{x}-\frac{8649}{100}\left(\frac{93}{10}\right)^{x}}{\frac{1000}{183^{3}}\times \left(\frac{183}{10}\right)^{x}+\frac{393}{10}\left(\frac{393}{10}\right)^{x}}\) - step7: Rewrite the expression: \(\frac{\frac{100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}}{100}}{\frac{1000}{183^{3}}\times \left(\frac{183}{10}\right)^{x}+\frac{393}{10}\left(\frac{393}{10}\right)^{x}}\) - step8: Rewrite the expression: \(\frac{\frac{100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}}{100}}{\frac{10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}}{10\times 183^{3}}}\) - step9: Multiply by the reciprocal: \(\frac{100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}}{100}\times \frac{10\times 183^{3}}{10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}}\) - step10: Reduce the fraction: \(\frac{100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}}{10}\times \frac{183^{3}}{10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}}\) - step11: Multiply the terms: \(\frac{\left(100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}\right)\times 183^{3}}{10\left(10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}\right)}\) - step12: Multiply the terms: \(\frac{183^{3}\left(100\left(\frac{193}{10}\right)^{x}-8649\left(\frac{93}{10}\right)^{x}\right)}{10\left(10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}\right)}\) - step13: Calculate: \(\frac{100\times 183^{3}\left(\frac{193}{10}\right)^{x}-8649\times 183^{3}\left(\frac{93}{10}\right)^{x}}{10\left(10000\left(\frac{183}{10}\right)^{x}+393\times 183^{3}\left(\frac{393}{10}\right)^{x}\right)}\) - step14: Calculate: \(\frac{100\times 183^{3}\left(\frac{193}{10}\right)^{x}-8649\times 183^{3}\left(\frac{93}{10}\right)^{x}}{100000\left(\frac{183}{10}\right)^{x}+3930\times 183^{3}\left(\frac{393}{10}\right)^{x}}\) - step15: Expand the expression: \(\frac{\frac{100\times 183^{3}\times 193^{x}-183^{3}\times 93^{x+2}}{10^{x}}}{10^{1-x}\left(10000\times 183^{x}+61^{3}\times 3^{4+x}\times 131^{x+1}\right)}\) - step16: Multiply by the reciprocal: \(\frac{100\times 183^{3}\times 193^{x}-183^{3}\times 93^{x+2}}{10^{x}}\times \frac{1}{10^{1-x}\left(10000\times 183^{x}+61^{3}\times 3^{4+x}\times 131^{x+1}\right)}\) - step17: Rewrite the expression: \(\frac{27\left(100\times 61^{3}\times 193^{x}-61^{3}\times 93^{x+2}\right)}{10^{x}}\times \frac{1}{10^{1-x}\times 27\left(10000\times 3^{x-3}\times 61^{x}+61^{3}\times 393^{x+1}\right)}\) - step18: Reduce the fraction: \(\frac{100\times 61^{3}\times 193^{x}-61^{3}\times 93^{x+2}}{10^{x}}\times \frac{1}{10^{1-x}\left(10000\times 3^{x-3}\times 61^{x}+61^{3}\times 393^{x+1}\right)}\) - step19: Rewrite the expression: \(\frac{61^{3}\left(100\times 193^{x}-93^{x+2}\right)}{10^{x}}\times \frac{1}{10^{1-x}\times 61^{3}\left(10000\times 183^{x-3}+393^{x+1}\right)}\) - step20: Reduce the fraction: \(\frac{100\times 193^{x}-93^{x+2}}{10^{x}}\times \frac{1}{10^{1-x}\left(10000\times 183^{x-3}+393^{x+1}\right)}\) - step21: Multiply the terms: \(\frac{100\times 193^{x}-93^{x+2}}{10^{x}\times 10^{1-x}\left(10000\times 183^{x-3}+393^{x+1}\right)}\) - step22: Multiply the terms: \(\frac{100\times 193^{x}-93^{x+2}}{100000\times 183^{x-3}+10\times 393^{x+1}}\) Let's simplify the given expressions step by step. ### 1.1.1 Simplifying \( \frac{19.3^{x}-9.3^{x+2}}{18.3^{x-3}+39.3^{1+x}} \) 1. **Factor out common terms in the numerator:** \[ 19.3^{x} - 9.3^{x+2} = 19.3^{x} - 9 \cdot 3^{2} \cdot 3^{x} = 19.3^{x} - 9 \cdot 9 \cdot 3^{x} = 19.3^{x} - 9 \cdot 9 \cdot 3^{x} \] This can be factored as: \[ 3^{x}(19 - 9 \cdot 9) = 3^{x}(19 - 81) = 3^{x}(-62) \] 2. **Factor out common terms in the denominator:** \[ 18.3^{x-3} + 39.3^{1+x} = 18 \cdot 3^{x-3} + 39 \cdot 3^{x+1} = 18 \cdot 3^{x-3} + 39 \cdot 3^{x} \cdot 3 \] This can be factored as: \[ 3^{x-3}(18 + 39 \cdot 3^{4}) = 3^{x-3}(18 + 117) = 3^{x-3}(135) \] 3. **Putting it all together:** \[ \frac{3^{x}(-62)}{3^{x-3}(135)} = \frac{-62 \cdot 3^{3}}{135} = \frac{-62 \cdot 27}{135} \] 4. **Final simplification:** \[ \frac{-1674}{135} = \frac{-62}{5} \text{ (after reducing)} \] Thus, the simplified form is: \[ \frac{100 \times 193^{x} - 93^{x+2}}{100000 \times 183^{x-3} + 10 \times 393^{x+1}} \] ### 1.1.2 Simplifying \( \frac{\sqrt{48 x^{6}}-\sqrt[3]{125 x^{9}} \cdot 3^{\frac{1}{2}}}{\sqrt{75 x^{6}}} \) 1. **Calculate each term:** - \( \sqrt{48 x^{6}} = \sqrt{48} \cdot x^{3} = 4\sqrt{3} \cdot x^{3} \) - \( \sqrt[3]{125 x^{9}} = 5x^{3} \) - Thus, \( \sqrt[3]{125 x^{9}} \cdot 3^{\frac{1}{2}} = 5x^{3} \cdot \sqrt{3} \) 2. **Combine the terms:** \[ \frac{4\sqrt{3} x^{3} - 5x^{3} \sqrt{3}}{\sqrt{75 x^{6}}} \] The numerator simplifies to: \[ (4 - 5)x^{3}\sqrt{3} = -x^{3}\sqrt{3} \] 3. **Denominator:** \[ \sqrt{75 x^{6}} = \sqrt{75} \cdot x^{3} = 5\sqrt{3} \cdot x^{3} \] 4. **Final expression:** \[ \frac{-x^{3}\sqrt{3}}{5\sqrt{3} \cdot x^{3}} = \frac{-1}{5} \] Thus, the simplified form is: \[ -\frac{1}{5} \text{ or } -0.2 \] ### Summary of Results: 1.1.1: \( \frac{100 \times 193^{x} - 93^{x+2}}{100000 \times 183^{x-3} + 10 \times 393^{x+1}} \) 1.1.2: \( -\frac{1}{5} \) or \( -0.2 \)