Initial program 30.1
\[\sqrt{x + 1} - \sqrt{x}\]
Initial simplification30.1
\[\leadsto \sqrt{1 + x} - \sqrt{x}\]
- Using strategy
rm Applied flip--29.9
\[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}}\]
- Using strategy
rm Applied *-un-lft-identity29.9
\[\leadsto \frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \color{blue}{1 \cdot \sqrt{x}}}\]
Applied *-un-lft-identity29.9
\[\leadsto \frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{1 \cdot \sqrt{1 + x}} + 1 \cdot \sqrt{x}}\]
Applied distribute-lft-out29.9
\[\leadsto \frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}}\]
Applied add-cube-cbrt29.9
\[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}} \cdot \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}\right) \cdot \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}}}{1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\]
Applied times-frac29.9
\[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}} \cdot \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}}{1} \cdot \frac{\sqrt[3]{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} + \sqrt{x}}}\]
Simplified29.9
\[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{1 + x} + \sqrt{x}}\]
Simplified0.2
\[\leadsto 1 \cdot \color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}\]
Final simplification0.2
\[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
