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
\[\leadsto \frac{\color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) - \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1 \cdot 1\right)}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Applied associate-/l/20.3
\[\leadsto \color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) - \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1 \cdot 1\right)}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)\right)}}\]
Simplified5.8
\[\leadsto \frac{\color{blue}{1}}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)\right)}\]
- Using strategy
rm Applied pow15.8
\[\leadsto \frac{1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}^{1}}\right)}\]
Applied pow15.8
\[\leadsto \frac{1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\left(\sqrt{x} \cdot \color{blue}{{\left(\sqrt{x}\right)}^{1}}\right) \cdot {\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}^{1}\right)}\]
Applied pow15.8
\[\leadsto \frac{1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\left(\color{blue}{{\left(\sqrt{x}\right)}^{1}} \cdot {\left(\sqrt{x}\right)}^{1}\right) \cdot {\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}^{1}\right)}\]
Applied pow-prod-down5.8
\[\leadsto \frac{1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\color{blue}{{\left(\sqrt{x} \cdot \sqrt{x}\right)}^{1}} \cdot {\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}^{1}\right)}\]
Applied pow-prod-down5.8
\[\leadsto \frac{1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \color{blue}{{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)\right)}^{1}}}\]
Applied pow15.8
\[\leadsto \frac{1}{\color{blue}{{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}^{1}} \cdot {\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)\right)}^{1}}\]
Applied pow-prod-down5.8
\[\leadsto \frac{1}{\color{blue}{{\left(\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)\right)\right)}^{1}}}\]
Simplified0.7
\[\leadsto \frac{1}{{\color{blue}{\left(x \cdot \left(\sqrt{x + 1} + \frac{x + 1}{\sqrt{x}}\right)\right)}}^{1}}\]
- Using strategy
rm Applied unpow-prod-down0.7
\[\leadsto \frac{1}{\color{blue}{{x}^{1} \cdot {\left(\sqrt{x + 1} + \frac{x + 1}{\sqrt{x}}\right)}^{1}}}\]
Applied associate-/r*0.4
\[\leadsto \color{blue}{\frac{\frac{1}{{x}^{1}}}{{\left(\sqrt{x + 1} + \frac{x + 1}{\sqrt{x}}\right)}^{1}}}\]
Simplified0.4
\[\leadsto \frac{\frac{1}{{x}^{1}}}{\color{blue}{\sqrt{1 + x} + \frac{1 + x}{\sqrt{x}}}}\]
Final simplification0.4
\[\leadsto \frac{\frac{1}{x}}{\sqrt{x + 1} + \frac{x + 1}{\sqrt{x}}}\]
