\( \sqrt { \frac { 5 ^ { 2 - x } - 4.5 ^ { - x } } { 5 ^ { - x } + 2.5 ^ { - x + 1 } } } \)

Calculate or simplify the expression \( \sqrt{\frac{5^{2-x}-4.5^{-x}}{5^{-x}+2.5^{-x+1}}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{\frac{5^{2-x}-4.5^{-x}}{5^{-x}+2.5^{-x+1}}}\) - step1: Convert the expressions: \(\sqrt{\frac{5^{2-x}-\left(\frac{9}{2}\right)^{-x}}{5^{-x}+2.5^{-x+1}}}\) - step2: Convert the expressions: \(\sqrt{\frac{5^{2-x}-\left(\frac{9}{2}\right)^{-x}}{5^{-x}+\left(\frac{5}{2}\right)^{-x+1}}}\) - step3: Divide the terms: \(\sqrt{\frac{50\left(\frac{45}{4}\right)^{x}-2\left(\frac{25}{2}\right)^{x}}{2\left(\frac{45}{4}\right)^{x}+5^{x+1}\left(\frac{9}{2}\right)^{x}}}\) - step4: Use the properties of radicals: \(\frac{\sqrt{50\left(\frac{45}{4}\right)^{x}-2\left(\frac{25}{2}\right)^{x}}}{\sqrt{2\left(\frac{45}{4}\right)^{x}+5^{x+1}\left(\frac{9}{2}\right)^{x}}}\) - step5: Simplify: \(\frac{\sqrt{100\left(\frac{45}{4}\right)^{2x}+50\left(\frac{405}{8}\right)^{x}\times 5^{x+1}-4\left(\frac{1125}{8}\right)^{x}-2\left(\frac{225}{4}\right)^{x}\times 5^{x+1}}}{2\left(\frac{45}{4}\right)^{x}+5^{x+1}\left(\frac{9}{2}\right)^{x}}\) - step6: Expand the expression: \(\frac{\frac{\sqrt{5^{2x+2}\times 9^{2x}+5\times 5^{2x+2}\times 9^{2x}\times 2^{x-1}-5^{3x}\times 9^{x}\times 2^{x}-5^{3x+1}\times 9^{x}\times 2^{2x-1}}}{2^{2x-1}}}{\frac{2^{1-x}\times 45^{x}+5^{x+1}\times 9^{x}}{2^{x}}}\) - step7: Multiply by the reciprocal: \(\frac{\sqrt{5^{2x+2}\times 9^{2x}+5\times 5^{2x+2}\times 9^{2x}\times 2^{x-1}-5^{3x}\times 9^{x}\times 2^{x}-5^{3x+1}\times 9^{x}\times 2^{2x-1}}}{2^{2x-1}}\times \frac{2^{x}}{2^{1-x}\times 45^{x}+5^{x+1}\times 9^{x}}\) - step8: Rewrite the expression: \(\frac{\sqrt{25\times 9^{x}+125\times 2^{x-1}\times 9^{x}-10^{x}-5^{x+1}\times 2^{2x-1}}\times 3^{x}\times 5^{x}}{2^{2x-1}}\times \frac{2^{x}}{5^{x}\times 3^{x}\left(3^{x}\times 2^{1-x}+5\times 3^{x}\right)}\) - step9: Reduce the fraction: \(\frac{\sqrt{25\times 9^{x}+125\times 2^{x-1}\times 9^{x}-10^{x}-5^{x+1}\times 2^{2x-1}}}{2^{x-1}}\times \frac{1}{3^{x}\times 2^{1-x}+5\times 3^{x}}\) - step10: Multiply the terms: \(\frac{\sqrt{25\times 9^{x}+125\times 2^{x-1}\times 9^{x}-10^{x}-5^{x+1}\times 2^{2x-1}}}{2^{x-1}\left(3^{x}\times 2^{1-x}+5\times 3^{x}\right)}\) The simplified form of the expression \( \sqrt{\frac{5^{2-x}-4.5^{-x}}{5^{-x}+2.5^{-x+1}}} \) is \( \frac{\sqrt{25 \times 9^{x} + 125 \times 2^{x-1} \times 9^{x} - 10^{x} - 5^{x+1} \times 2^{2x-1}}}{2^{x-1} (3^{x} \times 2^{1-x} + 5 \times 3^{x})} \).