Initial program 19.9
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
Simplified19.9
\[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot \left(y + x\right)}{(x \cdot x + \left(y \cdot y\right))_*}}\]
- Using strategy
rm Applied add-sqr-sqrt19.9
\[\leadsto \frac{\left(x - y\right) \cdot \left(y + x\right)}{\color{blue}{\sqrt{(x \cdot x + \left(y \cdot y\right))_*} \cdot \sqrt{(x \cdot x + \left(y \cdot y\right))_*}}}\]
Applied times-frac19.9
\[\leadsto \color{blue}{\frac{x - y}{\sqrt{(x \cdot x + \left(y \cdot y\right))_*}} \cdot \frac{y + x}{\sqrt{(x \cdot x + \left(y \cdot y\right))_*}}}\]
Simplified19.9
\[\leadsto \color{blue}{\frac{x - y}{\sqrt{x^2 + y^2}^*}} \cdot \frac{y + x}{\sqrt{(x \cdot x + \left(y \cdot y\right))_*}}\]
Simplified0.0
\[\leadsto \frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \color{blue}{\frac{x + y}{\sqrt{x^2 + y^2}^*}}\]
- Using strategy
rm Applied add-cbrt-cube31.4
\[\leadsto \frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \frac{x + y}{\color{blue}{\sqrt[3]{\left(\sqrt{x^2 + y^2}^* \cdot \sqrt{x^2 + y^2}^*\right) \cdot \sqrt{x^2 + y^2}^*}}}\]
Applied add-cbrt-cube31.4
\[\leadsto \frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \frac{\color{blue}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}{\sqrt[3]{\left(\sqrt{x^2 + y^2}^* \cdot \sqrt{x^2 + y^2}^*\right) \cdot \sqrt{x^2 + y^2}^*}}\]
Applied cbrt-undiv31.4
\[\leadsto \frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \color{blue}{\sqrt[3]{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}{\left(\sqrt{x^2 + y^2}^* \cdot \sqrt{x^2 + y^2}^*\right) \cdot \sqrt{x^2 + y^2}^*}}}\]
Applied add-cbrt-cube31.4
\[\leadsto \color{blue}{\sqrt[3]{\left(\frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \frac{x - y}{\sqrt{x^2 + y^2}^*}\right) \cdot \frac{x - y}{\sqrt{x^2 + y^2}^*}}} \cdot \sqrt[3]{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}{\left(\sqrt{x^2 + y^2}^* \cdot \sqrt{x^2 + y^2}^*\right) \cdot \sqrt{x^2 + y^2}^*}}\]
Applied cbrt-unprod31.4
\[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \frac{x - y}{\sqrt{x^2 + y^2}^*}\right) \cdot \frac{x - y}{\sqrt{x^2 + y^2}^*}\right) \cdot \frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}{\left(\sqrt{x^2 + y^2}^* \cdot \sqrt{x^2 + y^2}^*\right) \cdot \sqrt{x^2 + y^2}^*}}}\]
Simplified0.0
\[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{y + x}{\sqrt{x^2 + y^2}^*}\right)}^{3} \cdot {\left(\frac{x - y}{\sqrt{x^2 + y^2}^*}\right)}^{3}}}\]
- Using strategy
rm Applied cube-mult0.0
\[\leadsto \sqrt[3]{\color{blue}{\left(\frac{y + x}{\sqrt{x^2 + y^2}^*} \cdot \left(\frac{y + x}{\sqrt{x^2 + y^2}^*} \cdot \frac{y + x}{\sqrt{x^2 + y^2}^*}\right)\right)} \cdot {\left(\frac{x - y}{\sqrt{x^2 + y^2}^*}\right)}^{3}}\]
Applied associate-*l*0.0
\[\leadsto \sqrt[3]{\color{blue}{\frac{y + x}{\sqrt{x^2 + y^2}^*} \cdot \left(\left(\frac{y + x}{\sqrt{x^2 + y^2}^*} \cdot \frac{y + x}{\sqrt{x^2 + y^2}^*}\right) \cdot {\left(\frac{x - y}{\sqrt{x^2 + y^2}^*}\right)}^{3}\right)}}\]
Final simplification0.0
\[\leadsto \sqrt[3]{\frac{x + y}{\sqrt{x^2 + y^2}^*} \cdot \left({\left(\frac{x - y}{\sqrt{x^2 + y^2}^*}\right)}^{3} \cdot \left(\frac{x + y}{\sqrt{x^2 + y^2}^*} \cdot \frac{x + y}{\sqrt{x^2 + y^2}^*}\right)\right)}\]
