Initial program 19.8
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied flip--19.9
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
Applied simplify19.9
\[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
- Using strategy
rm Applied frac-sub19.2
\[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(x + 1\right) - x \cdot 1}{x \cdot \left(x + 1\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
Applied associate-/l/19.2
\[\leadsto \color{blue}{\frac{1 \cdot \left(x + 1\right) - x \cdot 1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(x \cdot \left(x + 1\right)\right)}}\]
- Using strategy
rm Applied add-sqr-sqrt19.3
\[\leadsto \color{blue}{\sqrt{\frac{1 \cdot \left(x + 1\right) - x \cdot 1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(x \cdot \left(x + 1\right)\right)}} \cdot \sqrt{\frac{1 \cdot \left(x + 1\right) - x \cdot 1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(x \cdot \left(x + 1\right)\right)}}}\]
Applied simplify19.3
\[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}}}} \cdot \sqrt{\frac{1 \cdot \left(x + 1\right) - x \cdot 1}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(x \cdot \left(x + 1\right)\right)}}\]
Applied simplify0.6
\[\leadsto \sqrt{\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}}} \cdot \color{blue}{\sqrt{\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}}}}\]
- Using strategy
rm Applied pow1/20.6
\[\leadsto \sqrt{\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}}} \cdot \color{blue}{{\left(\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}}\right)}^{\frac{1}{2}}}\]
Applied pow1/20.6
\[\leadsto \color{blue}{{\left(\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}}\right)}^{\frac{1}{2}}\]
Applied pow-prod-up0.4
\[\leadsto \color{blue}{{\left(\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}}\right)}^{\left(\frac{1}{2} + \frac{1}{2}\right)}}\]
