Initial program 25.5
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification25.5
\[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt25.5
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity25.5
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac25.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified25.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified16.3
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied div-sub16.3
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(\frac{b \cdot c}{\sqrt{d^2 + c^2}^*} - \frac{a \cdot d}{\sqrt{d^2 + c^2}^*}\right)}\]
- Using strategy
rm Applied *-un-lft-identity16.3
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(\frac{b \cdot c}{\sqrt{d^2 + c^2}^*} - \frac{a \cdot d}{\color{blue}{1 \cdot \sqrt{d^2 + c^2}^*}}\right)\]
Applied times-frac9.0
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(\frac{b \cdot c}{\sqrt{d^2 + c^2}^*} - \color{blue}{\frac{a}{1} \cdot \frac{d}{\sqrt{d^2 + c^2}^*}}\right)\]
Applied add-sqr-sqrt9.1
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(\frac{b \cdot c}{\color{blue}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}} - \frac{a}{1} \cdot \frac{d}{\sqrt{d^2 + c^2}^*}\right)\]
Applied times-frac0.8
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(\color{blue}{\frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}} \cdot \frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}}} - \frac{a}{1} \cdot \frac{d}{\sqrt{d^2 + c^2}^*}\right)\]
Applied prod-diff0.8
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left((\left(\frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}}\right) \cdot \left(\frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}}\right) + \left(-\frac{d}{\sqrt{d^2 + c^2}^*} \cdot \frac{a}{1}\right))_* + (\left(-\frac{d}{\sqrt{d^2 + c^2}^*}\right) \cdot \left(\frac{a}{1}\right) + \left(\frac{d}{\sqrt{d^2 + c^2}^*} \cdot \frac{a}{1}\right))_*\right)}\]
Applied distribute-rgt-in0.8
\[\leadsto \color{blue}{(\left(\frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}}\right) \cdot \left(\frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}}\right) + \left(-\frac{d}{\sqrt{d^2 + c^2}^*} \cdot \frac{a}{1}\right))_* \cdot \frac{1}{\sqrt{d^2 + c^2}^*} + (\left(-\frac{d}{\sqrt{d^2 + c^2}^*}\right) \cdot \left(\frac{a}{1}\right) + \left(\frac{d}{\sqrt{d^2 + c^2}^*} \cdot \frac{a}{1}\right))_* \cdot \frac{1}{\sqrt{d^2 + c^2}^*}}\]
Simplified0.8
\[\leadsto (\left(\frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}}\right) \cdot \left(\frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}}\right) + \left(-\frac{d}{\sqrt{d^2 + c^2}^*} \cdot \frac{a}{1}\right))_* \cdot \frac{1}{\sqrt{d^2 + c^2}^*} + \color{blue}{0}\]
Final simplification0.8
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot (\left(\frac{b}{\sqrt{\sqrt{d^2 + c^2}^*}}\right) \cdot \left(\frac{c}{\sqrt{\sqrt{d^2 + c^2}^*}}\right) + \left(\left(-a\right) \cdot \frac{d}{\sqrt{d^2 + c^2}^*}\right))_*\]
