Initial program 20.1
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Initial simplification20.1
\[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied frac-sub20.0
\[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
Simplified20.0
\[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
- Using strategy
rm Applied flip--19.8
\[\leadsto \frac{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
Applied associate-/l/19.8
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]
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
\[\leadsto \frac{1}{{\color{blue}{\left((x \cdot \left(\sqrt{x + 1}\right) + \left((\left(\sqrt{x}\right) \cdot x + \left(\sqrt{x}\right))_*\right))_*\right)}}^{1}}\]
- Using strategy
rm Applied add-cube-cbrt0.7
\[\leadsto \frac{1}{{\left((x \cdot \left(\sqrt{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}\right) + \left((\left(\sqrt{x}\right) \cdot x + \left(\sqrt{x}\right))_*\right))_*\right)}^{1}}\]
Applied sqrt-prod0.7
\[\leadsto \frac{1}{{\left((x \cdot \color{blue}{\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}\right)} + \left((\left(\sqrt{x}\right) \cdot x + \left(\sqrt{x}\right))_*\right))_*\right)}^{1}}\]
Simplified0.7
\[\leadsto \frac{1}{{\left((x \cdot \left(\color{blue}{\left|\sqrt[3]{1 + x}\right|} \cdot \sqrt{\sqrt[3]{x + 1}}\right) + \left((\left(\sqrt{x}\right) \cdot x + \left(\sqrt{x}\right))_*\right))_*\right)}^{1}}\]
Final simplification0.7
\[\leadsto \frac{1}{(x \cdot \left(\sqrt{\sqrt[3]{x + 1}} \cdot \left|\sqrt[3]{x + 1}\right|\right) + \left((\left(\sqrt{x}\right) \cdot x + \left(\sqrt{x}\right))_*\right))_*}\]
