Initial program 20.5
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied flip--20.5
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
- Using strategy
rm Applied frac-times25.5
\[\leadsto \frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Applied frac-times20.6
\[\leadsto \frac{\color{blue}{\frac{1 \cdot 1}{\sqrt{x} \cdot \sqrt{x}}} - \frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Applied frac-sub20.3
\[\leadsto \frac{\color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) - \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1 \cdot 1\right)}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Simplified5.9
\[\leadsto \frac{\frac{\color{blue}{1}}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Simplified5.7
\[\leadsto \frac{\frac{1}{\color{blue}{(x \cdot x + x)_*}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
- Using strategy
rm Applied *-un-lft-identity5.7
\[\leadsto \frac{\frac{1}{(x \cdot x + x)_*}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{\color{blue}{1 \cdot \left(x + 1\right)}}}}\]
Applied sqrt-prod5.7
\[\leadsto \frac{\frac{1}{(x \cdot x + x)_*}}{\frac{1}{\sqrt{x}} + \frac{1}{\color{blue}{\sqrt{1} \cdot \sqrt{x + 1}}}}\]
Applied *-un-lft-identity5.7
\[\leadsto \frac{\frac{1}{(x \cdot x + x)_*}}{\frac{1}{\sqrt{x}} + \frac{\color{blue}{1 \cdot 1}}{\sqrt{1} \cdot \sqrt{x + 1}}}\]
Applied times-frac5.7
\[\leadsto \frac{\frac{1}{(x \cdot x + x)_*}}{\frac{1}{\sqrt{x}} + \color{blue}{\frac{1}{\sqrt{1}} \cdot \frac{1}{\sqrt{x + 1}}}}\]
Applied *-un-lft-identity5.7
\[\leadsto \frac{\frac{1}{(x \cdot x + x)_*}}{\frac{1}{\sqrt{\color{blue}{1 \cdot x}}} + \frac{1}{\sqrt{1}} \cdot \frac{1}{\sqrt{x + 1}}}\]
Applied sqrt-prod5.7
\[\leadsto \frac{\frac{1}{(x \cdot x + x)_*}}{\frac{1}{\color{blue}{\sqrt{1} \cdot \sqrt{x}}} + \frac{1}{\sqrt{1}} \cdot \frac{1}{\sqrt{x + 1}}}\]
Applied *-un-lft-identity5.7
\[\leadsto \frac{\frac{1}{(x \cdot x + x)_*}}{\frac{\color{blue}{1 \cdot 1}}{\sqrt{1} \cdot \sqrt{x}} + \frac{1}{\sqrt{1}} \cdot \frac{1}{\sqrt{x + 1}}}\]
Applied times-frac5.7
\[\leadsto \frac{\frac{1}{(x \cdot x + x)_*}}{\color{blue}{\frac{1}{\sqrt{1}} \cdot \frac{1}{\sqrt{x}}} + \frac{1}{\sqrt{1}} \cdot \frac{1}{\sqrt{x + 1}}}\]
Applied distribute-lft-out5.7
\[\leadsto \frac{\frac{1}{(x \cdot x + x)_*}}{\color{blue}{\frac{1}{\sqrt{1}} \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}}\]
Applied *-un-lft-identity5.7
\[\leadsto \frac{\frac{1}{\color{blue}{1 \cdot (x \cdot x + x)_*}}}{\frac{1}{\sqrt{1}} \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}\]
Applied add-cube-cbrt5.7
\[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot (x \cdot x + x)_*}}{\frac{1}{\sqrt{1}} \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}\]
Applied times-frac5.7
\[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{(x \cdot x + x)_*}}}{\frac{1}{\sqrt{1}} \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}\]
Applied times-frac5.7
\[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{\frac{1}{\sqrt{1}}} \cdot \frac{\frac{\sqrt[3]{1}}{(x \cdot x + x)_*}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
Simplified5.7
\[\leadsto \color{blue}{1} \cdot \frac{\frac{\sqrt[3]{1}}{(x \cdot x + x)_*}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Simplified0.4
\[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{1 + x}}{\frac{x}{\sqrt{1 + x}} + \frac{x}{\sqrt{x}}}}\]
- Using strategy
rm Applied *-un-lft-identity0.4
\[\leadsto 1 \cdot \frac{\frac{1}{1 + x}}{\frac{x}{\sqrt{1 + x}} + \frac{x}{\color{blue}{1 \cdot \sqrt{x}}}}\]
Applied add-sqr-sqrt0.3
\[\leadsto 1 \cdot \frac{\frac{1}{1 + x}}{\frac{x}{\sqrt{1 + x}} + \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1 \cdot \sqrt{x}}}\]
Applied times-frac0.3
\[\leadsto 1 \cdot \frac{\frac{1}{1 + x}}{\frac{x}{\sqrt{1 + x}} + \color{blue}{\frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{\sqrt{x}}}}\]
Applied *-un-lft-identity0.3
\[\leadsto 1 \cdot \frac{\frac{1}{1 + x}}{\frac{x}{\color{blue}{1 \cdot \sqrt{1 + x}}} + \frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{\sqrt{x}}}\]
Applied add-sqr-sqrt0.3
\[\leadsto 1 \cdot \frac{\frac{1}{1 + x}}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1 \cdot \sqrt{1 + x}} + \frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{\sqrt{x}}}\]
Applied times-frac0.3
\[\leadsto 1 \cdot \frac{\frac{1}{1 + x}}{\color{blue}{\frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}} + \frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{\sqrt{x}}}\]
Applied distribute-lft-out0.3
\[\leadsto 1 \cdot \frac{\frac{1}{1 + x}}{\color{blue}{\frac{\sqrt{x}}{1} \cdot \left(\frac{\sqrt{x}}{\sqrt{1 + x}} + \frac{\sqrt{x}}{\sqrt{x}}\right)}}\]
Simplified0.3
\[\leadsto 1 \cdot \frac{\frac{1}{1 + x}}{\color{blue}{\sqrt{x}} \cdot \left(\frac{\sqrt{x}}{\sqrt{1 + x}} + \frac{\sqrt{x}}{\sqrt{x}}\right)}\]
Final simplification0.3
\[\leadsto \frac{\frac{1}{x + 1}}{\sqrt{x} \cdot \left(\frac{\sqrt{x}}{\sqrt{x}} + \frac{\sqrt{x}}{\sqrt{x + 1}}\right)}\]
