Initial program 14.0
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
Simplified12.6
\[\leadsto \color{blue}{\sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0}\]
- Using strategy
rm Applied add-cube-cbrt12.7
\[\leadsto \sqrt{1 - \left(\frac{h}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0\]
Applied *-un-lft-identity12.7
\[\leadsto \sqrt{1 - \left(\frac{\color{blue}{1 \cdot h}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0\]
Applied times-frac12.7
\[\leadsto \sqrt{1 - \left(\color{blue}{\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{h}{\sqrt[3]{\ell}}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0\]
Applied associate-*l*9.5
\[\leadsto \sqrt{1 - \color{blue}{\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{h}{\sqrt[3]{\ell}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right)} \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0\]
- Using strategy
rm Applied div-inv9.5
\[\leadsto \sqrt{1 - \left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\color{blue}{\left(h \cdot \frac{1}{\sqrt[3]{\ell}}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0\]
Applied associate-*l*8.3
\[\leadsto \sqrt{1 - \left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{\sqrt[3]{\ell}} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right)}\right) \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0\]
Final simplification8.3
\[\leadsto \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{\sqrt[3]{\ell}}\right) \cdot h\right)\right)} \cdot w0\]
