1. Solve the first expression step-by-step: \[ \left( (2.4+1)^{\frac{1}{5}} \cdot 4.375 - \left(\frac{2.75 - \frac{1}{5}}{2}\right)^{2} \right) \div \frac{67}{200} \]\nCalculate inside the parentheses: $2.4+1=3.4$ Take the fifth root: $3.4^{\frac{1}{5}}=\sqrt[5]{3.4} \approx 1.2804$ Multiply: $1.2804 \times 4.375 \approx 5.6018$ Calculate the fraction inside the square: $2.75 - \frac{1}{5} = 2.75 - 0.2 = 2.55$ Divide by 2: $\frac{2.55}{2} = 1.275$ Square it: $1.275^{2} = 1.6256$ Subtract: $5.6018 - 1.6256 = 3.9762$ Divide by $\frac{67}{200} = 0.335$: $\frac{3.9762}{0.335} = 11.8683$ Calculate second RHS expression: $8^{\frac{1}{20}} - 0.45$ $8^{\frac{1}{20}} = e^{\frac{1}{20} \ln 8} = e^{0.104} \approx 1.1098$ Subtract 0.45: $1.1098 - 0.45 = 0.6598$ 2. For the second problem: \[ \left( 6 - 4^{-\frac{1}{2}} \right)^{0.03} \div \left( 0.3 - 20^{-\frac{1}{5}} \right)^{\frac{1}{2}} = \frac{7}{50} \] Calculate inside parentheses: $4^{-\frac{1}{2}} = \frac{1}{\sqrt{4}} = 0.5$ $6 - 0.5 = 5.5$ Raise to 0.03 power: $5.5^{0.03} \approx e^{0.03 \ln 5.5} = e^{0.03 \times 1.7047} = e^{0.0511} = 1.0524$ Calculate denominator: $20^{-\frac{1}{5}} = e^{-\frac{1}{5} \ln 20} = e^{-0.2996} = 0.7415$ $0.3 - 0.7415 = -0.4415$ Square root of a negative number is undefined in real numbers, so expression is undefined or complex. So LHS is complex; RHS $\frac{7}{50} = 0.14$ real. No equality. Calculate second expression in problem 2: \[ \left( \frac{3}{2} - 2.65^{\frac{1}{4}} \right)^4 \div \left( \left( 1.88 + 2^{\frac{3}{5}} \right) \times 80 \right) \] Calculate $2.65^{\frac{1}{4}} = \sqrt[4]{2.65} \approx 1.2755$ $\frac{3}{2} - 1.2755 = 1.5 - 1.2755=0.2245$ Raise to power 4: $0.2245^4 = 0.00254$ Calculate denominator: $2^{\frac{3}{5}} = e^{\frac{3}{5} \ln 2} = e^{0.4159} = 1.5157$ Sum inside parentheses: $1.88 +1.5157 = 3.3957$ Multiply by 80: $3.3957 \times 80 = 271.656$ Divide numerator by denominator: $\frac{0.00254}{271.656} \approx 9.35 \times 10^{-6}$ 3. Third problem: \[ \left( 13^{\frac{1}{4}} \times 23 - 40^{\frac{1}{4}} \times 49 \right) \left[ \left( 4 - 3 \frac{1}{2} \left( 2 \frac{1}{7} - 1 \frac{1}{5} \right) \right)^{0.16} \right] \] Calculate roots: $13^{\frac{1}{4}} \approx \sqrt[4]{13} = 1.898$ $13^{\frac{1}{4}} \times 23 = 1.898 \times 23 = 43.65$ $40^{\frac{1}{4}} = \sqrt[4]{40} = 2.512$ $2.512 \times 49 = 123.09$ Subtract: $43.65 - 123.09 = -79.44$ Calculate inside bracket: $2 \frac{1}{7} = 2 + \frac{1}{7} = 2.1429$ $1 \frac{1}{5} = 1 + \frac{1}{5} = 1.2$ Difference: $2.1429 - 1.2 = 0.9429$ Multiply by $3 \frac{1}{2} = 3.5$ $3.5 \times 0.9429 = 3.300$ Subtract from 4: $4 - 3.300 = 0.7$ Raise to $0.16$ power: $0.7^{0.16} = e^{0.16 \ln 0.7} = e^{0.16 \times (-0.3567)} = e^{-0.0571} = 0.9445$ Multiply entire expression: $-79.44 \times 0.9445 = -75.02$ 4. Fourth problem: \[ \left( 46^{\frac{2}{3}} \times 12 + 41^{\frac{1}{35}} \times 23 \div 260 \right) 5 + 800 \times \left( 12 \frac{28}{31} \right) \] Calculate: $46^{\frac{2}{3}} = (46^{1/3})^2 = (\sqrt[3]{46})^2$ $\sqrt[3]{46} \approx 3.57$ Square: $3.57^2 = 12.74$ Multiply by 12: $12.74 \times 12 = 152.88$ Calculate $41^{\frac{1}{35}} = e^{\frac{1}{35} \ln 41} = e^{0.1119} = 1.1185$ Multiply by 23: $1.1185 \times 23 = 25.71$ Divide by 260: $25.71 / 260 = 0.0989$ Sum inside parentheses: $152.88 + 0.0989 = 152.98$ Multiply by 5: $152.98 \times 5 = 764.9$ Convert mixed number $12 \frac{28}{31} = 12 + \frac{28}{31} = 12.9032$ Multiply by 800: $12.9032 \times 800 = 10322.56$ Sum total: $764.9 + 10322.56 = 11087.46$ Calculate second expression: $0.8 \cdot 7.2 \cdot 4.5 \cdot 1.35 = 0.8 \times 7.2 = 5.76$ $5.76 \times 4.5 = 25.92$ $25.92 \times 1.35 = 35.00$ 5. Fifth problem: Calculate: $6.5 \times 2.7 \times 1.92^{\frac{1}{8}}$ Calculate $1.92^{\frac{1}{8}} = e^{\frac{1}{8} \ln 1.92} = e^{0.083 / 8} = e^{0.0104} = 1.0105$ Multiply: $6.5 \times 2.7 = 17.55$ $17.55 \times 1.0105 = 17.74$ Second expression: $0.8 \times 1.25^{\frac{1}{4}} \times (1.08 - \frac{2}{25})^2 \div (6^{-\frac{5}{9}} - \frac{3}{2})^2 + (1.2 \times 0.5) \times \frac{4}{5}$ Calculate: $1.25^{\frac{1}{4}} = \sqrt[4]{1.25} = 1.0574$ $1.08 - \frac{2}{25} = 1.08 - 0.08 = 1.0$ Square: $1.0^2 = 1$ Calculate $6^{-\frac{5}{9}} = e^{- \frac{5}{9} \ln 6} = e^{-0.892} = 0.409$ $0.409 - 1.5 = -1.091$ Square: $(-1.091)^2 = 1.19$ Divide: $0.8 \times 1.0574 \times 1 / 1.19 = 0.711$ Calculate $(1.2 \times 0.5) \times \frac{4}{5} = 0.6 \times 0.8 = 0.48$ Sum: $0.711 + 0.48 = 1.191$ 6. Sixth problem: \[ \left[ 41 \frac{29}{72} \div \left(18 \frac{7}{8} - 5^{\frac{1}{4}}\right) \times \left( 10 \frac{1}{2} - 7 \frac{3}{8} \right) \right] \div 22 \frac{7}{18} \] Convert mixed numbers: $41 \frac{29}{72} = 41.4028$ $18 \frac{7}{8} = 18.875$ Calculate $5^{\frac{1}{4}} = \sqrt[4]{5} = 1.495$ Difference: $18.875 - 1.495 = 17.38$ $10 \frac{1}{2} = 10.5$ $7 \frac{3}{8} = 7.375$ Difference: $10.5 - 7.375 = 3.125$ Calculate numerator bracket: $41.4028 \div 17.38 = 2.382$ Multiply by $3.125 = 7.44$ Denominator: $22 \frac{7}{18} = 22.389$ Divide: $7.44 \div 22.389 = 0.3325$ Calculate: \[ (1.75^2 - 1.75^{\frac{1}{8}})^2 \] Calculate powers: $1.75^2=3.0625$ $1.75^{\frac{1}{8}} = e^{\frac{1}{8}\ln 1.75} = e^{0.0713} = 1.074$ Difference: $3.0625 - 1.074 = 1.9885$ Square: $1.9885^2 = 3.954$ \[ \left( \frac{17}{80} - 0.0325 \right) \div 400 \] Calculate fraction: $\frac{17}{80} = 0.2125$ Difference: $0.2125 - 0.0325 = 0.18$ Divide by 400: $0.18 / 400 = 0.00045$ Finally: \[ (6.79 : 0.7 + 0.3) = \frac{6.79}{0.7} + 0.3 = 9.7 + 0.3 = 10.0 \] 8. Problem 8: \[ \left( \frac{0.216}{0.15} + \frac{2 \cdot 4}{3} \div 15 \right) + \left( \frac{196}{225} - \frac{7.7}{24} \right) + 0.695 : 1.39 = \] Calculate first bracket: $\frac{0.216}{0.15} = 1.44$ $\frac{2 \cdot 4}{3} = \frac{8}{3} = 2.6667$ Divide by 15: $2.6667 / 15 = 0.1778$ Sum: $1.44 + 0.1778=1.6178$ Second bracket: $\frac{196}{225} = 0.8711$ $\frac{7.7}{24} = 0.3208$ Difference: $0.8711 - 0.3208 = 0.5503$ Calculate 0.695 divided by 1.39: $0.695 / 1.39 = 0.5$ Sum all: $1.6178 + 0.5503 + 0.5 = 2.6681$ 9. Problem 9: \[ \left[ \left( 6 - 4^{-\frac{1}{2}} \right)^{0.03} \div \left( 0.3 - 20^{-\frac{1}{5}} \right)^{\frac{1}{2}} \right] \cdot 62 \frac{1}{20} \] Previously computed that denominator is negative, undefined in real numbers, so problem is complex. Mixed number: $62 \frac{1}{20} = 62.05$ Second expression: \[ \left( 3 \frac{1}{20} - 2.65 \right)^{\frac{1}{4}} \cdot \left[ \left( 1.88 + 2^{\frac{3}{5}} \right) \cdot 8 \right] \cdot 4.5 \div \left[ 38.375 - \left( 26^{\frac{1}{8}} - 18 \cdot 0.75 \right) : 2.4 \cdot 0.8 \right] \] $3 \frac{1}{20} = 3.05$ Difference: $3.05 - 2.65 = 0.4$ Root: $0.4^{\frac{1}{4}} = \sqrt[4]{0.4} = 0.796$ Calculate $2^{\frac{3}{5}} = 1.5157$ Sum with 1.88: $1.88 + 1.5157 = 3.3957$ Multiply by 8: $3.3957 \times 8 = 27.166$ Multiply by 4.5: $27.166 \times 4.5 = 122.25$ Calculate $26^{\frac{1}{8}} = e^{\frac{1}{8} \ln 26} = e^{0.386} = 1.471$ Calculate inside brackets: $18 \cdot 0.75 = 13.5$ Difference: $1.471 - 13.5 = -12.029$ Divide by 2.4: $-12.029 / 2.4 = -5.012$ Multiply by 0.8: $-5.012 \times 0.8 = -4.01$ Calculate denominator: $38.375 - (-4.01) = 38.375 + 4.01 = 42.385$ Divide numerator by denominator: $122.25 / 42.385 = 2.884$ Third expression: \[17.81 : 1.37 - 2^{-\frac{23}{31}} \cdot 16\] Divide: $17.81 / 1.37 = 13.0$ Calculate power: $2^{-\frac{23}{31}} = e^{-\frac{23}{31} \ln 2} = e^{-0.513} = 0.598$ Multiply: $0.598 \times 16 = 9.57$ Subtract: $13.0 - 9.57 = 3.43$ Final answers summarized: 1) $11.87 \approx 0.66$ (not equal) 2) Undefined / complex; second part approximately $9.35 \times 10^{-6}$ 3) $-75.02$ 4) $11087.46$ and $35.00$ 5) $17.74$ and $1.191$ 6) $0.333$, $3.954$, $0.00045$, $10.0$ 8) $2.67$ 9) Complex first part, $2.88$ second part, $3.43$ third part