Initial program 19.5
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied flip--19.5
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
- Using strategy
rm Applied frac-times24.5
\[\leadsto \frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Applied frac-times19.6
\[\leadsto \frac{\color{blue}{\frac{1 \cdot 1}{\sqrt{x} \cdot \sqrt{x}}} - \frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Applied frac-sub19.4
\[\leadsto \frac{\color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) - \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1 \cdot 1\right)}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Simplified5.6
\[\leadsto \frac{\frac{\color{blue}{1}}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Simplified5.5
\[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
- Using strategy
rm Applied *-un-lft-identity5.5
\[\leadsto \frac{\frac{1}{x \cdot x + x}}{\frac{1}{\sqrt{x}} + \color{blue}{1 \cdot \frac{1}{\sqrt{x + 1}}}}\]
Applied div-inv5.5
\[\leadsto \frac{\frac{1}{x \cdot x + x}}{\color{blue}{1 \cdot \frac{1}{\sqrt{x}}} + 1 \cdot \frac{1}{\sqrt{x + 1}}}\]
Applied distribute-lft-out5.5
\[\leadsto \frac{\frac{1}{x \cdot x + x}}{\color{blue}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}}\]
Applied div-inv5.5
\[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{x \cdot x + x}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}\]
Applied times-frac5.5
\[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{\frac{1}{x \cdot x + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
Simplified5.5
\[\leadsto \color{blue}{1} \cdot \frac{\frac{1}{x \cdot x + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Simplified0.8
\[\leadsto 1 \cdot \color{blue}{\frac{1}{\left(\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}\right) \cdot \left(x + 1\right)}}\]
- Using strategy
rm Applied associate-/r*0.4
\[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}}}{x + 1}}\]
Final simplification0.4
\[\leadsto \frac{\frac{1}{\frac{x}{\sqrt{x}} + \frac{x}{\sqrt{x + 1}}}}{x + 1}\]
