Initial program 19.4
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Initial simplification19.4
\[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied add-sqr-sqrt24.4
\[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\sqrt{\frac{1}{\sqrt{x + 1}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1}}}}\]
Applied add-sqr-sqrt19.6
\[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x}}}} - \sqrt{\frac{1}{\sqrt{x + 1}}} \cdot \sqrt{\frac{1}{\sqrt{x + 1}}}\]
Applied difference-of-squares19.6
\[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\sqrt{x}}} + \sqrt{\frac{1}{\sqrt{x + 1}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt{x}}} - \sqrt{\frac{1}{\sqrt{x + 1}}}\right)}\]
- Using strategy
rm Applied sqrt-div22.6
\[\leadsto \left(\sqrt{\frac{1}{\sqrt{x}}} + \sqrt{\frac{1}{\sqrt{x + 1}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt{x}}} - \color{blue}{\frac{\sqrt{1}}{\sqrt{\sqrt{x + 1}}}}\right)\]
Applied sqrt-div19.6
\[\leadsto \left(\sqrt{\frac{1}{\sqrt{x}}} + \sqrt{\frac{1}{\sqrt{x + 1}}}\right) \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\sqrt{x}}}} - \frac{\sqrt{1}}{\sqrt{\sqrt{x + 1}}}\right)\]
Applied frac-sub19.6
\[\leadsto \left(\sqrt{\frac{1}{\sqrt{x}}} + \sqrt{\frac{1}{\sqrt{x + 1}}}\right) \cdot \color{blue}{\frac{\sqrt{1} \cdot \sqrt{\sqrt{x + 1}} - \sqrt{\sqrt{x}} \cdot \sqrt{1}}{\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}}}\]
Applied sqrt-div19.6
\[\leadsto \left(\sqrt{\frac{1}{\sqrt{x}}} + \color{blue}{\frac{\sqrt{1}}{\sqrt{\sqrt{x + 1}}}}\right) \cdot \frac{\sqrt{1} \cdot \sqrt{\sqrt{x + 1}} - \sqrt{\sqrt{x}} \cdot \sqrt{1}}{\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}}\]
Applied sqrt-div19.7
\[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\sqrt{x}}}} + \frac{\sqrt{1}}{\sqrt{\sqrt{x + 1}}}\right) \cdot \frac{\sqrt{1} \cdot \sqrt{\sqrt{x + 1}} - \sqrt{\sqrt{x}} \cdot \sqrt{1}}{\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}}\]
Applied frac-add19.7
\[\leadsto \color{blue}{\frac{\sqrt{1} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{\sqrt{x}} \cdot \sqrt{1}}{\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}}} \cdot \frac{\sqrt{1} \cdot \sqrt{\sqrt{x + 1}} - \sqrt{\sqrt{x}} \cdot \sqrt{1}}{\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}}\]
Applied frac-times19.6
\[\leadsto \color{blue}{\frac{\left(\sqrt{1} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{\sqrt{x}} \cdot \sqrt{1}\right) \cdot \left(\sqrt{1} \cdot \sqrt{\sqrt{x + 1}} - \sqrt{\sqrt{x}} \cdot \sqrt{1}\right)}{\left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}\right)}}\]
Simplified19.6
\[\leadsto \frac{\color{blue}{\left(\sqrt{\sqrt{1 + x}} + \sqrt{\sqrt{x}}\right) \cdot \left(\sqrt{\sqrt{1 + x}} - \sqrt{\sqrt{x}}\right)}}{\left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}\right)}\]
Simplified19.5
\[\leadsto \frac{\left(\sqrt{\sqrt{1 + x}} + \sqrt{\sqrt{x}}\right) \cdot \left(\sqrt{\sqrt{1 + x}} - \sqrt{\sqrt{x}}\right)}{\color{blue}{\sqrt{x} \cdot \sqrt{1 + x}}}\]
- Using strategy
rm Applied flip3--19.5
\[\leadsto \frac{\left(\sqrt{\sqrt{1 + x}} + \sqrt{\sqrt{x}}\right) \cdot \color{blue}{\frac{{\left(\sqrt{\sqrt{1 + x}}\right)}^{3} - {\left(\sqrt{\sqrt{x}}\right)}^{3}}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}} + \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}} + \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{x}}\right)}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
Applied flip3-+19.5
\[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt{\sqrt{1 + x}}\right)}^{3} + {\left(\sqrt{\sqrt{x}}\right)}^{3}}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}} + \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}} - \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{x}}\right)}} \cdot \frac{{\left(\sqrt{\sqrt{1 + x}}\right)}^{3} - {\left(\sqrt{\sqrt{x}}\right)}^{3}}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}} + \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}} + \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{x}}\right)}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
Applied frac-times19.5
\[\leadsto \frac{\color{blue}{\frac{\left({\left(\sqrt{\sqrt{1 + x}}\right)}^{3} + {\left(\sqrt{\sqrt{x}}\right)}^{3}\right) \cdot \left({\left(\sqrt{\sqrt{1 + x}}\right)}^{3} - {\left(\sqrt{\sqrt{x}}\right)}^{3}\right)}{\left(\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}} + \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}} - \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{x}}\right)\right) \cdot \left(\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}} + \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}} + \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{x}}\right)\right)}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
Applied associate-/l/19.5
\[\leadsto \color{blue}{\frac{\left({\left(\sqrt{\sqrt{1 + x}}\right)}^{3} + {\left(\sqrt{\sqrt{x}}\right)}^{3}\right) \cdot \left({\left(\sqrt{\sqrt{1 + x}}\right)}^{3} - {\left(\sqrt{\sqrt{x}}\right)}^{3}\right)}{\left(\sqrt{x} \cdot \sqrt{1 + x}\right) \cdot \left(\left(\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}} + \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}} - \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{x}}\right)\right) \cdot \left(\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}} + \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}} + \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{x}}\right)\right)\right)}}\]
Final simplification19.5
\[\leadsto \frac{\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{3} + {\left(\sqrt{\sqrt{x}}\right)}^{3}\right) \cdot \left({\left(\sqrt{\sqrt{x + 1}}\right)}^{3} - {\left(\sqrt{\sqrt{x}}\right)}^{3}\right)}{\left(\sqrt{x + 1} \cdot \sqrt{x}\right) \cdot \left(\left(\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}} + \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x}}\right)\right) \cdot \left(\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}} - \sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x}}\right)\right)\right)}\]
