\(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\sqrt[3]{x}}\right)\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
13.1
- Using strategy
rm 13.1
- Applied add-exp-log to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}}\]
13.1
- Using strategy
rm 13.1
- Applied add-sqr-sqrt to get
\[e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}}\right)} \leadsto e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2}\right)}\]
13.3
- Applied add-sqr-sqrt to get
\[e^{\log \left(\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)} \leadsto e^{\log \left(\color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}\]
13.3
- Applied difference-of-squares to get
\[e^{\log \color{red}{\left({\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}} \leadsto e^{\log \color{blue}{\left(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)\right)}}\]
13.3
- Applied log-prod to get
\[e^{\color{red}{\log \left(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)\right)}} \leadsto e^{\color{blue}{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) + \log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}}\]
13.4
- Applied taylor to get
\[e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) + \log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)} \leadsto e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) + \log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\frac{1}{3}}}\right)}\]
13.4
- Taylor expanded around 0 to get
\[e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) + \log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\color{red}{{x}^{\frac{1}{3}}}}\right)} \leadsto e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) + \log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\color{blue}{{x}^{\frac{1}{3}}}}\right)}\]
13.4
- Applied simplify to get
\[e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) + \log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\frac{1}{3}}}\right)} \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\sqrt[3]{x}}\right)\]
13.0
- Applied final simplification
