Initial program 20.3
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied frac-sub20.3
\[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
Simplified20.3
\[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
- Using strategy
rm Applied flip--20.1
\[\leadsto \frac{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
Applied associate-/l/20.1
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]
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 distribute-lft-in0.8
\[\leadsto \frac{1}{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x}}}\]
Simplified0.8
\[\leadsto \frac{1}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1} + \color{blue}{\sqrt{x + 1} \cdot x}}\]
Final simplification0.8
\[\leadsto \frac{1}{x \cdot \sqrt{x + 1} + \sqrt{x + 1} \cdot \left(\sqrt{x + 1} \cdot \sqrt{x}\right)}\]
