Initial program 19.1
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied frac-sub19.1
\[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
Simplified19.1
\[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
- Using strategy
rm Applied flip--18.9
\[\leadsto \frac{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
Applied associate-/l/18.9
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]
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 pow10.8
\[\leadsto \frac{1}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{1}}}\]
Applied pow10.8
\[\leadsto \frac{1}{\left(\sqrt{x} \cdot \color{blue}{{\left(\sqrt{x + 1}\right)}^{1}}\right) \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{1}}\]
Applied pow10.8
\[\leadsto \frac{1}{\left(\color{blue}{{\left(\sqrt{x}\right)}^{1}} \cdot {\left(\sqrt{x + 1}\right)}^{1}\right) \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{1}}\]
Applied pow-prod-down0.8
\[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{x} \cdot \sqrt{x + 1}\right)}^{1}} \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{1}}\]
Applied pow-prod-down0.8
\[\leadsto \frac{1}{\color{blue}{{\left(\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{1}}}\]
Simplified0.7
\[\leadsto \frac{1}{{\color{blue}{\left((x \cdot \left(\sqrt{x + 1}\right) + \left((\left(\sqrt{x}\right) \cdot x + \left(\sqrt{x}\right))_*\right))_*\right)}}^{1}}\]
Final simplification0.7
\[\leadsto \frac{1}{(x \cdot \left(\sqrt{x + 1}\right) + \left((\left(\sqrt{x}\right) \cdot x + \left(\sqrt{x}\right))_*\right))_*}\]
