\(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot {\left(\log \left(e^{\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\right)}^3\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
14.6
- Using strategy
rm 14.6
- Applied add-sqr-sqrt to get
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}\]
15.0
- Applied add-sqr-sqrt to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2 \leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2\]
14.7
- Applied difference-of-squares to get
\[\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2} \leadsto \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
14.7
- Using strategy
rm 14.7
- Applied add-cube-cbrt to get
\[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{red}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)} \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}^3}\]
14.7
- Using strategy
rm 14.7
- Applied add-log-exp to get
\[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot {\color{red}{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}}^3 \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot {\color{blue}{\left(\log \left(e^{\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\right)}}^3\]
14.8
