Initial program 19.4
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Initial simplification19.4
\[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- 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--19.3
\[\leadsto \frac{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
Applied associate-/l/19.3
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]
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 associate-/r*0.4
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1} + \sqrt{x}}}\]
- Using strategy
rm Applied add-cube-cbrt0.5
\[\leadsto \frac{\frac{1}{\sqrt{x} \cdot \sqrt{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}}}{\sqrt{x + 1} + \sqrt{x}}\]
Applied sqrt-prod0.5
\[\leadsto \frac{\frac{1}{\sqrt{x} \cdot \color{blue}{\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}\right)}}}{\sqrt{x + 1} + \sqrt{x}}\]
Applied associate-*r*0.4
\[\leadsto \frac{\frac{1}{\color{blue}{\left(\sqrt{x} \cdot \sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \sqrt{\sqrt[3]{x + 1}}}}}{\sqrt{x + 1} + \sqrt{x}}\]
Final simplification0.4
\[\leadsto \frac{\frac{1}{\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{x}\right) \cdot \sqrt{\sqrt[3]{x + 1}}}}{\sqrt{x} + \sqrt{x + 1}}\]
