Initial program 30.0
\[\sqrt{x + 1} - \sqrt{x}\]
- Using strategy
rm Applied flip--29.8
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\]
- Using strategy
rm Applied add-sqr-sqrt29.8
\[\leadsto \frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{\sqrt{x + 1} + \sqrt{x}} \cdot \sqrt{\sqrt{x + 1} + \sqrt{x}}}}\]
Applied add-cube-cbrt29.9
\[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}} \cdot \sqrt[3]{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}\right) \cdot \sqrt[3]{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}} \cdot \sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
Applied times-frac29.9
\[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}} \cdot \sqrt[3]{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\sqrt[3]{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}\]
Simplified29.8
\[\leadsto \color{blue}{\frac{{\left(\left(1 + x\right) - x\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}}} \cdot \frac{\sqrt[3]{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
Simplified29.4
\[\leadsto \frac{{\left(\left(1 + x\right) - x\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}} \cdot \color{blue}{\frac{\sqrt[3]{x + \left(1 - x\right)}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}}}\]
- Using strategy
rm Applied pow129.4
\[\leadsto \frac{{\left(\left(1 + x\right) - x\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}} \cdot \color{blue}{{\left(\frac{\sqrt[3]{x + \left(1 - x\right)}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}}\right)}^{1}}\]
Applied pow129.4
\[\leadsto \color{blue}{{\left(\frac{{\left(\left(1 + x\right) - x\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}}\right)}^{1}} \cdot {\left(\frac{\sqrt[3]{x + \left(1 - x\right)}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}}\right)}^{1}\]
Applied pow-prod-down29.4
\[\leadsto \color{blue}{{\left(\frac{{\left(\left(1 + x\right) - x\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}} \cdot \frac{\sqrt[3]{x + \left(1 - x\right)}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}}\right)}^{1}}\]
Simplified0.2
\[\leadsto {\color{blue}{\left(\frac{1 - 0}{\sqrt{x + 1} + \sqrt{x}}\right)}}^{1}\]
Final simplification0.2
\[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}}\]
