Initial program 40.4
\[\frac{e^{x}}{e^{x} - 1}\]
Initial simplification0.5
\[\leadsto \frac{e^{x}}{(e^{x} - 1)^*}\]
- Using strategy
rm Applied *-un-lft-identity0.5
\[\leadsto \frac{e^{x}}{\color{blue}{1 \cdot (e^{x} - 1)^*}}\]
Applied add-sqr-sqrt0.5
\[\leadsto \frac{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}}{1 \cdot (e^{x} - 1)^*}\]
Applied times-frac0.5
\[\leadsto \color{blue}{\frac{\sqrt{e^{x}}}{1} \cdot \frac{\sqrt{e^{x}}}{(e^{x} - 1)^*}}\]
Simplified0.5
\[\leadsto \color{blue}{\sqrt{e^{x}}} \cdot \frac{\sqrt{e^{x}}}{(e^{x} - 1)^*}\]
- Using strategy
rm Applied add-cube-cbrt0.5
\[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}}} \cdot \frac{\sqrt{e^{x}}}{(e^{x} - 1)^*}\]
Applied sqrt-prod0.5
\[\leadsto \color{blue}{\left(\sqrt{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt{\sqrt[3]{e^{x}}}\right)} \cdot \frac{\sqrt{e^{x}}}{(e^{x} - 1)^*}\]
Applied associate-*l*0.5
\[\leadsto \color{blue}{\sqrt{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \left(\sqrt{\sqrt[3]{e^{x}}} \cdot \frac{\sqrt{e^{x}}}{(e^{x} - 1)^*}\right)}\]
Simplified0.5
\[\leadsto \color{blue}{\left|\sqrt[3]{e^{x}}\right|} \cdot \left(\sqrt{\sqrt[3]{e^{x}}} \cdot \frac{\sqrt{e^{x}}}{(e^{x} - 1)^*}\right)\]
Final simplification0.5
\[\leadsto \left(\frac{\sqrt{e^{x}}}{(e^{x} - 1)^*} \cdot \sqrt{\sqrt[3]{e^{x}}}\right) \cdot \left|\sqrt[3]{e^{x}}\right|\]
