Math Problem Solver

$$\frac{1.35}{3} \left(31.56 + 22.08 + \sqrt{31.56} \cdot 22.08\right) = \frac{12069}{500} + \frac{1242 \sqrt{789}}{625} \approx 79.95674657997974070954138208777699$$

$$\begin{aligned}\frac{1.35}{3} \left(31.56 + 22.08 + \sqrt{31.56} \cdot 22.08\right)&=\left(1 + \frac{35}{100}\right) \cdot \frac{1}{3} \cdot \left(31 + \frac{56}{100} + 22 + \frac{8}{100} + \sqrt{31 + \frac{56}{100}} \cdot \left(22 + \frac{8}{100}\right)\right)\\&=\left(1 + \frac{7}{20}\right) \cdot \frac{1}{3} \cdot \left(31 + \frac{56}{100} + 22 + \frac{8}{100} + \sqrt{31 + \frac{56}{100}} \cdot \left(22 + \frac{8}{100}\right)\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(31 + \frac{56}{100} + 22 + \frac{8}{100} + \sqrt{31 + \frac{56}{100}} \cdot \left(22 + \frac{8}{100}\right)\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(31 + \frac{14}{25} + 22 + \frac{8}{100} + \sqrt{31 + \frac{56}{100}} \cdot \left(22 + \frac{8}{100}\right)\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(31 + \frac{14}{25} + 22 + \frac{2}{25} + \sqrt{31 + \frac{56}{100}} \cdot \left(22 + \frac{8}{100}\right)\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(31 + \frac{14}{25} + 22 + \frac{2}{25} + \sqrt{31 + \frac{14}{25}} \cdot \left(22 + \frac{8}{100}\right)\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(31 + \frac{14}{25} + 22 + \frac{2}{25} + \sqrt{\frac{789}{25}} \cdot \left(22 + \frac{8}{100}\right)\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(31 + \frac{14}{25} + 22 + \frac{2}{25} + \frac{1}{5} \cdot \sqrt{789} \cdot \left(22 + \frac{8}{100}\right)\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(31 + \frac{14}{25} + 22 + \frac{2}{25} + \frac{1}{5} \cdot \sqrt{789} \cdot \left(22 + \frac{2}{25}\right)\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(31 + \frac{14}{25} + 22 + \frac{2}{25} + \frac{1}{5} \cdot \sqrt{789} \cdot \frac{552}{25}\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(31 + \frac{14}{25} + 22 + \frac{2}{25} + \frac{552}{125} \cdot \sqrt{789}\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(\frac{789}{25} + 22 + \frac{2}{25} + \frac{552}{125} \cdot \sqrt{789}\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(\frac{1339}{25} + \frac{2}{25} + \frac{552}{125} \cdot \sqrt{789}\right)\\&=\frac{27}{20} \cdot \frac{1}{3} \cdot \left(\frac{1341}{25} + \frac{552}{125} \cdot \sqrt{789}\right)\\&=\frac{9}{20} \cdot \left(\frac{1341}{25} + \frac{552}{125} \cdot \sqrt{789}\right)\\&=\frac{12069}{500} + \frac{1242}{625} \cdot \sqrt{789}\end{aligned}$$