Solve y=(2+x^3)^2

$$y = ( 2 + x ^ { 3 } ) ^ { 2 }$$

Solve for y

$$y=\left(x^{3}+2\right)^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+x^{3}\right)^{2}.

$$y=4+4x^{3}+\left(x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$y=4+4x^{3}+x^{6}$$

Solve for x (complex solution)

$$x\in \sqrt[3]{\sqrt{y}-2},e^{\frac{4\pi i}{3}}\sqrt[3]{\sqrt{y}-2},e^{\frac{2\pi i}{3}}\sqrt[3]{\sqrt{y}-2},e^{\frac{5\pi i}{3}}\sqrt[3]{\sqrt{y}+2},e^{\frac{\pi i}{3}}\sqrt[3]{\sqrt{y}+2},-\sqrt[3]{\sqrt{y}+2}$$

Solve for x

$$x=-\sqrt[3]{\sqrt{y}+2}$$

$$x=\sqrt[3]{\sqrt{y}-2}\text{, }y\geq 0$$