Initial program 19.1
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied frac-sub19.1
\[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
Simplified19.1
\[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
- Using strategy
rm Applied flip--18.9
\[\leadsto \frac{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
Applied associate-/l/18.9
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]
Simplified0.8
\[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]
- Using strategy
rm Applied associate-/r*0.4
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1} + \sqrt{x}}}\]
- Using strategy
rm Applied flip3-+0.8
\[\leadsto \frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\color{blue}{\frac{{\left(\sqrt{x + 1}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)}}}\]
Final simplification0.8
\[\leadsto \frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\frac{{\left(\sqrt{x + 1}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)}}\]
