\(\frac{1}{\left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot \sqrt[3]{x} + {\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}^2}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
29.5
- Using strategy
rm 29.5
- Applied flip3-- to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
29.4
- Applied simplify to get
\[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
29.4
- Applied taylor to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
29.4
- Taylor expanded around 0 to get
\[\frac{\color{red}{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
29.4
- Applied simplify to get
\[\color{red}{\frac{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}} \leadsto \color{blue}{\frac{1}{\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) \cdot {x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}}}\]
3.0
- Applied taylor to get
\[\frac{1}{\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) \cdot {x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}} \leadsto \frac{1}{\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) \cdot {x}^{\frac{1}{3}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}}\]
3.0
- Taylor expanded around 0 to get
\[\frac{1}{\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) \cdot \color{red}{{x}^{\frac{1}{3}}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}} \leadsto \frac{1}{\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) \cdot \color{blue}{{x}^{\frac{1}{3}}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}}\]
3.0
- Applied simplify to get
\[\frac{1}{\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) \cdot {x}^{\frac{1}{3}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}} \leadsto \frac{1}{\left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot \sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}}\]
2.7
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{1}{\left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot \sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}}} \leadsto \color{blue}{\frac{1}{\left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot \sqrt[3]{x} + {\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}^2}}\]
2.7
