\(\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{(\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right) * \left(\sqrt[3]{x} + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right) + \left({\left(\sqrt[3]{x}\right)}^2\right))_*}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
29.8
- Using strategy
rm 29.8
- Applied flip3-- to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
29.7
- Applied simplify to get
\[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
29.7
- Applied taylor to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\frac{1}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
29.7
- Taylor expanded around 0 to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\color{red}{\left({x}^{\frac{1}{3}}\right)}}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\color{blue}{\left({x}^{\frac{1}{3}}\right)}}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
29.7
- Applied simplify to get
\[\color{red}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\frac{1}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{(\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right) * \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right))_*}}\]
29.7
- Applied taylor to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{(\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right) * \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right))_*} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{(\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right) * \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} + {x}^{\frac{1}{3}}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right))_*}\]
29.7
- Taylor expanded around 0 to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{(\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right) * \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} + \color{red}{{x}^{\frac{1}{3}}}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right))_*} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{(\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right) * \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} + \color{blue}{{x}^{\frac{1}{3}}}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right))_*}\]
29.7
- Applied simplify to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{(\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right) * \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} + {x}^{\frac{1}{3}}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right))_*} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{(\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right) * \left(\sqrt[3]{x} + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right) + \left({\left(\sqrt[3]{x}\right)}^2\right))_*}\]
29.7
- Applied final simplification
