Math Problem Solver

\[\frac{{15.24}^{0.02} {\left( \frac{1}{74} \right)}^{0.88}}{{15.24}^{0.02} {\left( \frac{1}{74} \right)}^{0.88} + {22.16}^{0.02} {\left( \frac{1}{92} \right)}^{0.88} + {35.38}^{0.02} {\left( \frac{1}{34} \right)}^{0.88}} = \frac{\sqrt[50]{381} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}}}{370 \cdot \left(\frac{2^{\frac{13}{50}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277} \cdot 5^{\frac{24}{25}}}{460} + \frac{\sqrt[50]{381} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}}}{370} + \frac{17^{\frac{3}{25}} \cdot \sqrt[50]{1769} \cdot \sqrt[10]{2} \cdot 5^{\frac{24}{25}}}{170}\right)} \approx 0.25987088011604274193002616232858\]

\[\begin{aligned}\frac{{15.24}^{0.02} {\left( \frac{1}{74} \right)}^{0.88}}{{15.24}^{0.02} {\left( \frac{1}{74} \right)}^{0.88} + {22.16}^{0.02} {\left( \frac{1}{92} \right)}^{0.88} + {35.38}^{0.02} {\left( \frac{1}{34} \right)}^{0.88}}& = \left(15 + \frac{24}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} \cdot \frac{1}{\left(15 + \frac{24}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \left(15 + \frac{6}{25}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} \cdot \frac{1}{\left(15 + \frac{24}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \left(\frac{381}{25}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} \cdot \frac{1}{\left(15 + \frac{24}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \sqrt[50]{\frac{381}{25}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} \cdot \frac{1}{\left(15 + \frac{24}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} \cdot \frac{1}{\left(15 + \frac{24}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \left(\frac{1}{74}\right)^{\frac{22}{25}} \cdot \frac{1}{\left(15 + \frac{24}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\left(15 + \frac{24}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\left(15 + \frac{6}{25}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\left(\frac{381}{25}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\sqrt[50]{\frac{381}{25}} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \left(\frac{1}{74}\right)^{\frac{88}{100}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \left(\frac{1}{74}\right)^{\frac{22}{25}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \left(22 + \frac{16}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \left(22 + \frac{4}{25}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \left(\frac{554}{25}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \sqrt[50]{\frac{554}{25}} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{554} \cdot \left(\frac{1}{92}\right)^{\frac{88}{100}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{554} \cdot \left(\frac{1}{92}\right)^{\frac{22}{25}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{554} \cdot \frac{1}{92} \cdot 92^{\frac{3}{25}} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277} + \left(35 + \frac{38}{100}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277} + \left(35 + \frac{19}{50}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277} + \left(\frac{1769}{50}\right)^{\frac{2}{100}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277} + \sqrt[50]{\frac{1769}{50}} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277} + \frac{1}{10} \cdot 2^{\frac{49}{50}} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{1769} \cdot \left(\frac{1}{34}\right)^{\frac{88}{100}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277} + \frac{1}{10} \cdot 2^{\frac{49}{50}} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{1769} \cdot \left(\frac{1}{34}\right)^{\frac{22}{25}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277} + \frac{1}{10} \cdot 2^{\frac{49}{50}} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{1769} \cdot \frac{1}{34} \cdot 34^{\frac{3}{25}}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277} + \frac{1}{170} \cdot \sqrt[10]{2} \cdot 5^{\frac{24}{25}} \cdot 17^{\frac{3}{25}} \cdot \sqrt[50]{1769}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\left(\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277}\right) + \frac{1}{170} \cdot \sqrt[10]{2} \cdot 5^{\frac{24}{25}} \cdot 17^{\frac{3}{25}} \cdot \sqrt[50]{1769}}\\& = \frac{1}{5} \cdot 5^{\frac{24}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{74} \cdot 74^{\frac{3}{25}} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{170} \cdot \sqrt[10]{2} \cdot 5^{\frac{24}{25}} \cdot 17^{\frac{3}{25}} \cdot \sqrt[50]{1769} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277}}\\& = \frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} \cdot \frac{1}{\frac{1}{370} \cdot 5^{\frac{24}{25}} \cdot 74^{\frac{3}{25}} \cdot \sqrt[50]{381} + \frac{1}{170} \cdot \sqrt[10]{2} \cdot 5^{\frac{24}{25}} \cdot 17^{\frac{3}{25}} \cdot \sqrt[50]{1769} + \frac{1}{460} \cdot 2^{\frac{13}{50}} \cdot 5^{\frac{24}{25}} \cdot 23^{\frac{3}{25}} \cdot \sqrt[50]{277}}\end{aligned}\]