Initial program 30.7
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
Simplified0.6
\[\leadsto \color{blue}{\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log 10}}\]
- Using strategy
rm Applied add-sqr-sqrt0.6
\[\leadsto \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]
Applied pow10.6
\[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re^2 + im^2}^*\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
Applied log-pow0.6
\[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re^2 + im^2}^*\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]
Applied times-frac0.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\sqrt{\log 10}}}\]
- Using strategy
rm Applied div-inv0.4
\[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\sqrt{re^2 + im^2}^*\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]
Applied associate-*r*0.4
\[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re^2 + im^2}^*\right)\right) \cdot \frac{1}{\sqrt{\log 10}}}\]
- Using strategy
rm Applied add-cube-cbrt0.4
\[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\]
Applied log-prod0.5
\[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) + \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\]
Applied distribute-lft-in0.4
\[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)\right)} \cdot \frac{1}{\sqrt{\log 10}}\]
- Using strategy
rm Applied pow1/30.4
\[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) + \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(\sqrt{re^2 + im^2}^*\right)}^{\frac{1}{3}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\]
Final simplification0.4
\[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) + \log \left({\left(\sqrt{re^2 + im^2}^*\right)}^{\frac{1}{3}}\right) \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\]
