\({\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)}^3\right)}^3\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
14.8
- Using strategy
rm 14.8
- Applied add-cube-cbrt to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
14.8
- Using strategy
rm 14.8
- Applied add-cube-cbrt to get
\[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3\]
14.8
- Using strategy
rm 14.8
- Applied add-sqr-sqrt to get
\[{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}}\right)}^3\right)}^3\]
15.2
- Applied add-sqr-sqrt to get
\[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)}^3\right)}^3\]
14.9
- Applied difference-of-squares to get
\[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}}\right)}^3\right)}^3\]
14.9
