\(\frac{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left({\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^{3} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^{3}\right)}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 + \left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2 + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
13.2
- Using strategy
rm 13.2
- Applied add-sqr-sqrt to get
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2}\]
13.4
- Applied add-sqr-sqrt to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2 \leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\]
13.3
- Applied difference-of-squares to get
\[\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2} \leadsto \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}\]
13.3
- Using strategy
rm 13.3
- Applied flip3-- to get
\[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \color{red}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)} \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \color{blue}{\frac{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^{3} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^{3}}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 + \left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2 + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}}\]
13.4
- Applied associate-*r/ to get
\[\color{red}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \frac{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^{3} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^{3}}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 + \left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2 + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}} \leadsto \color{blue}{\frac{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left({\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^{3} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^{3}\right)}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 + \left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2 + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}}\]
13.4
