Initial program 30.2
\[\sqrt{x + 1} - \sqrt{x}\]
- Using strategy
rm Applied add-sqr-sqrt30.2
\[\leadsto \sqrt{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}} - \sqrt{x}\]
Applied sqrt-prod30.3
\[\leadsto \color{blue}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}} - \sqrt{x}\]
- Using strategy
rm Applied flip--30.2
\[\leadsto \color{blue}{\frac{\left(\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}}\]
- Using strategy
rm Applied prod-diff30.3
\[\leadsto \frac{\color{blue}{(\left(\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}\right) + \left(-\sqrt{x} \cdot \sqrt{x}\right))_* + (\left(-\sqrt{x}\right) \cdot \left(\sqrt{x}\right) + \left(\sqrt{x} \cdot \sqrt{x}\right))_*}}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}\]
Simplified30.3
\[\leadsto \frac{\color{blue}{1} + (\left(-\sqrt{x}\right) \cdot \left(\sqrt{x}\right) + \left(\sqrt{x} \cdot \sqrt{x}\right))_*}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}\]
Simplified0.3
\[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}\]
- Using strategy
rm Applied pow1/20.3
\[\leadsto \frac{1 + 0}{\sqrt{\sqrt{x + 1}} \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{\frac{1}{2}}} + \sqrt{x}}\]
Applied pow1/20.3
\[\leadsto \frac{1 + 0}{\color{blue}{{\left(\sqrt{x + 1}\right)}^{\frac{1}{2}}} \cdot {\left(\sqrt{x + 1}\right)}^{\frac{1}{2}} + \sqrt{x}}\]
Applied pow-prod-up0.2
\[\leadsto \frac{1 + 0}{\color{blue}{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{2} + \frac{1}{2}\right)}} + \sqrt{x}}\]
Final simplification0.2
\[\leadsto \frac{1}{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{2} + \frac{1}{2}\right)} + \sqrt{x}}\]
