Initial program 19.4
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied frac-sub19.4
\[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
Simplified19.4
\[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
- Using strategy
rm Applied flip--19.1
\[\leadsto \frac{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
Applied associate-/l/19.1
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]
Simplified0.8
\[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]
- Using strategy
rm Applied *-un-lft-identity0.8
\[\leadsto \frac{\color{blue}{1 \cdot 1}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]
Applied times-frac0.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}\]
- Using strategy
rm Applied pow10.4
\[\leadsto \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \cdot \color{blue}{{\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{1}}\]
Applied pow10.4
\[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}\right)}^{1}} \cdot {\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{1}\]
Applied pow-prod-down0.4
\[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)}^{1}}\]
Simplified0.6
\[\leadsto {\color{blue}{\left(\frac{1}{\sqrt{x} \cdot \left(x + 1\right) + x \cdot \sqrt{x + 1}}\right)}}^{1}\]
Final simplification0.6
\[\leadsto \frac{1}{\sqrt{x + 1} \cdot x + \sqrt{x} \cdot \left(x + 1\right)}\]
