\(\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{e^{\frac{\log \left(x + 1\right)}{3}} \cdot \left(e^{\frac{\log \left(x + 1\right)}{3}} + \sqrt[3]{x}\right) + {x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
29.3
- Using strategy
rm 29.3
- Applied add-exp-log to get
\[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
29.5
- Applied pow-exp to get
\[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
29.5
- Applied simplify to get
\[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)} \leadsto e^{\color{blue}{\frac{\log \left(x + 1\right)}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
29.2
- Using strategy
rm 29.2
- Applied flip3-- to get
\[\color{red}{e^{\frac{\log \left(x + 1\right)}{3}} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
29.3
- Applied simplify to get
\[\frac{\color{red}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
29.2
- Applied taylor to get
\[\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\frac{1}{3}}\right)}\]
29.2
- Taylor expanded around 0 to get
\[\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot \color{red}{{x}^{\frac{1}{3}}}\right)} \leadsto \frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right)}\]
29.2
- Applied simplify to get
\[\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\frac{1}{3}}\right)} \leadsto \frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{\left({\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \sqrt[3]{x} \cdot e^{\frac{\log \left(x + 1\right)}{3}}\right) + {x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}}\]
29.2
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{\left({\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \sqrt[3]{x} \cdot e^{\frac{\log \left(x + 1\right)}{3}}\right) + {x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}}} \leadsto \color{blue}{\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{e^{\frac{\log \left(x + 1\right)}{3}} \cdot \left(e^{\frac{\log \left(x + 1\right)}{3}} + \sqrt[3]{x}\right) + {x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}}}\]
29.3
