Initial program 25.8
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification25.8
\[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt25.8
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity25.8
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac25.8
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified25.8
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified16.5
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*r/16.4
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(b \cdot c - a \cdot d\right)}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied sub-neg16.4
\[\leadsto \frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(b \cdot c + \left(-a \cdot d\right)\right)}}{\sqrt{d^2 + c^2}^*}\]
Applied distribute-rgt-in16.4
\[\leadsto \frac{\color{blue}{\left(b \cdot c\right) \cdot \frac{1}{\sqrt{d^2 + c^2}^*} + \left(-a \cdot d\right) \cdot \frac{1}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
Simplified9.0
\[\leadsto \frac{\left(b \cdot c\right) \cdot \frac{1}{\sqrt{d^2 + c^2}^*} + \color{blue}{a \cdot \frac{-d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
- Using strategy
rm Applied associate-*l*0.5
\[\leadsto \frac{\color{blue}{b \cdot \left(c \cdot \frac{1}{\sqrt{d^2 + c^2}^*}\right)} + a \cdot \frac{-d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
Final simplification0.5
\[\leadsto \frac{\left(-a\right) \cdot \frac{d}{\sqrt{d^2 + c^2}^*} + b \cdot \left(\frac{1}{\sqrt{d^2 + c^2}^*} \cdot c\right)}{\sqrt{d^2 + c^2}^*}\]
