Solve y=(1-2x^2)^3

$$y = ( 1 - 2 x ^ { 2 } ) ^ { 3 }$$

Solve for y

$$y=\left(1-2x^{2}\right)^{3}$$

Solution Steps

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

$$y=1-6x^{2}+12\left(x^{2}\right)^{2}-8\left(x^{2}\right)^{3}$$

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

$$y=1-6x^{2}+12x^{4}-8\left(x^{2}\right)^{3}$$

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

$$y=1-6x^{2}+12x^{4}-8x^{6}$$

Solve for x

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

$$x=-\frac{\sqrt{-2\sqrt[3]{y}+2}}{2}\text{, }y\leq 1$$