Initial program 19.7
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied flip--19.8
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
Applied simplify19.8
\[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
- Using strategy
rm Applied frac-sub19.2
\[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(x + 1\right) - x \cdot 1}{x \cdot \left(x + 1\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Applied simplify5.5
\[\leadsto \frac{\frac{\color{blue}{1}}{x \cdot \left(x + 1\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Applied simplify5.5
\[\leadsto \frac{\frac{1}{\color{blue}{(x \cdot x + x)_*}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
- Using strategy
rm Applied clear-num5.5
\[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}{\frac{1}{(x \cdot x + x)_*}}}}\]
- Using strategy
rm Applied add-sqr-sqrt5.7
\[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}{\frac{1}{(x \cdot x + x)_*}}}} \cdot \sqrt{\frac{1}{\frac{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}{\frac{1}{(x \cdot x + x)_*}}}}}\]
Applied simplify5.6
\[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x}} + \frac{x}{\sqrt{x + 1}}}}} \cdot \sqrt{\frac{1}{\frac{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}{\frac{1}{(x \cdot x + x)_*}}}}\]
Applied simplify0.6
\[\leadsto \sqrt{\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x}} + \frac{x}{\sqrt{x + 1}}}} \cdot \color{blue}{\sqrt{\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x}} + \frac{x}{\sqrt{x + 1}}}}}\]
