Solve v=(x^2+1)^3

$$v = ( x ^ { 2 } + 1 ) ^ { 3 }$$

Solve for v

$$v=\left(x^{2}+1\right)^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x^{2}+1\right)^{3}.

$$v=\left(x^{2}\right)^{3}+3\left(x^{2}\right)^{2}+3x^{2}+1$$

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

$$v=x^{6}+3\left(x^{2}\right)^{2}+3x^{2}+1$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$v=x^{6}+3x^{4}+3x^{2}+1$$

Solve for x (complex solution)

$$x\in i\sqrt{e^{\frac{\pi i}{3}}\sqrt[3]{v}+1},-i\sqrt{e^{\frac{\pi i}{3}}\sqrt[3]{v}+1},-\sqrt{\sqrt[3]{v}-1},\sqrt{\sqrt[3]{v}-1},-\sqrt{e^{\frac{2\pi i}{3}}\sqrt[3]{v}-1},\sqrt{e^{\frac{2\pi i}{3}}\sqrt[3]{v}-1}$$

Solve for x

$$x=\sqrt{\sqrt[3]{v}-1}$$

$$x=-\sqrt{\sqrt[3]{v}-1}\text{, }v\geq 1$$