\(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\sqrt[3]{x}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
13.6
- Using strategy
rm 13.6
- Applied add-log-exp to get
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right)}\]
14.8
- Applied add-log-exp to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right) \leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right)\]
14.4
- Applied diff-log to get
\[\color{red}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right) - \log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right)} \leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}}{e^{{x}^{\left(\frac{1}{3}\right)}}}\right)}\]
14.5
- Applied simplify to get
\[\log \color{red}{\left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}}{e^{{x}^{\left(\frac{1}{3}\right)}}}\right)} \leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}}\right)}\]
13.7
- Using strategy
rm 13.7
- Applied add-sqr-sqrt to get
\[\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}}}\right) \leadsto \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2}}\right)\]
14.0
- Applied add-sqr-sqrt to get
\[\log \left(e^{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2}\right) \leadsto \log \left(e^{\color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2}\right)\]
13.8
- Applied difference-of-squares to get
\[\log \left(e^{\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2}}\right) \leadsto \log \left(e^{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}}\right)\]
13.8
- Applied taylor to get
\[\log \left(e^{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}\right) \leadsto \log \left(e^{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\frac{1}{3}}}\right)}\right)\]
13.8
- Taylor expanded around 0 to get
\[\log \left(e^{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\color{red}{{x}^{\frac{1}{3}}}}\right)}\right) \leadsto \log \left(e^{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\color{blue}{{x}^{\frac{1}{3}}}}\right)}\right)\]
13.8
- Applied simplify to get
\[\log \left(e^{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\frac{1}{3}}}\right)}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\sqrt[3]{x}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)\]
13.5
- Applied final simplification
