Initial program 30.8
\[\sqrt{x + 1} - \sqrt{x}\]
Initial simplification30.8
\[\leadsto \sqrt{1 + x} - \sqrt{x}\]
- Using strategy
rm Applied flip--30.6
\[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}}\]
- Using strategy
rm Applied *-un-lft-identity30.6
\[\leadsto \frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}}\]
Applied add-sqr-sqrt30.6
\[\leadsto \frac{\color{blue}{\sqrt{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}} \cdot \sqrt{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}}}{1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\]
Applied times-frac30.6
\[\leadsto \color{blue}{\frac{\sqrt{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}}{1} \cdot \frac{\sqrt{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} + \sqrt{x}}}\]
Simplified30.5
\[\leadsto \color{blue}{1} \cdot \frac{\sqrt{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} + \sqrt{x}}\]
Simplified0.2
\[\leadsto 1 \cdot \color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\]
- Using strategy
rm Applied add-sqr-sqrt0.2
\[\leadsto 1 \cdot \frac{1}{\sqrt{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}} + \sqrt{x}}\]
Applied sqrt-prod0.3
\[\leadsto 1 \cdot \frac{1}{\color{blue}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}} + \sqrt{x}}\]
Applied fma-def0.2
\[\leadsto 1 \cdot \frac{1}{\color{blue}{(\left(\sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x + 1}}\right) + \left(\sqrt{x}\right))_*}}\]
Final simplification0.2
\[\leadsto \frac{1}{(\left(\sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x + 1}}\right) + \left(\sqrt{x}\right))_*}\]
