Initial program 29.9
\[\sqrt{x + 1} - \sqrt{x}\]
- Using strategy
rm Applied flip--29.7
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\]
Applied simplify0.2
\[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
- Using strategy
rm Applied add-sqr-sqrt0.3
\[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}\]
- Using strategy
rm Applied pow1/20.3
\[\leadsto \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \cdot \color{blue}{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{\frac{1}{2}}}\]
Applied pow1/20.3
\[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{\frac{1}{2}}\]
Applied pow-prod-down0.2
\[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{\frac{1}{2}}}\]
Applied simplify0.3
\[\leadsto {\color{blue}{\left(\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\right)}}^{\frac{1}{2}}\]
- Using strategy
rm Applied pow10.3
\[\leadsto {\left(\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{1}}}\right)}^{\frac{1}{2}}\]
Applied pow10.3
\[\leadsto {\left(\frac{1}{\color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{1}} \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{1}}\right)}^{\frac{1}{2}}\]
Applied pow-prod-up0.3
\[\leadsto {\left(\frac{1}{\color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(1 + 1\right)}}}\right)}^{\frac{1}{2}}\]
Applied pow-flip0.2
\[\leadsto {\color{blue}{\left({\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(-\left(1 + 1\right)\right)}\right)}}^{\frac{1}{2}}\]
