Initial program 20.1
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- 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
\[\leadsto \frac{1}{{\color{blue}{\left(x \cdot \sqrt{x + 1} + \left(x + 1\right) \cdot \sqrt{x}\right)}}^{1}}\]
- Using strategy
rm Applied add-cube-cbrt0.8
\[\leadsto \frac{1}{{\left(x \cdot \sqrt{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} + \left(x + 1\right) \cdot \sqrt{x}\right)}^{1}}\]
Applied sqrt-prod0.8
\[\leadsto \frac{1}{{\left(x \cdot \color{blue}{\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}\right)} + \left(x + 1\right) \cdot \sqrt{x}\right)}^{1}}\]
Applied associate-*r*0.8
\[\leadsto \frac{1}{{\left(\color{blue}{\left(x \cdot \sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \sqrt{\sqrt[3]{x + 1}}} + \left(x + 1\right) \cdot \sqrt{x}\right)}^{1}}\]
Final simplification0.8
\[\leadsto \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + \sqrt{\sqrt[3]{x + 1}} \cdot \left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot x\right)}\]
