Expand the expression using the formula:
[tex]a^3+b^3=(a+b)(a^2 - ab + b^2)[/tex]
[tex]64x^3+1=0 \\
(4x)^3+1^3=0 \\
(4x+1)((4x)^2- 4x \times 1+1^2)=0 \\
(4x+1)(16x^2-4x+1)=0 \\
4x+1=0 \ \lor \ 16x^2-4x+1=0[/tex]
The first equation:
[tex]4x+1=0 \\ 4x=-1 \\ x=-\frac{1}{4}[/tex]
The second equation:
[tex]16x^2-4x+1=0 \\ \\
a=16 \\ b=-4 \\ c=1 \\ b^2-4ac=(-4)^2-4 \times 16 \times 1=16-64=-48 \\
\sqrt{b^2-4ac}=\sqrt{-48}=\sqrt{48} \times \sqrt{-1}=\sqrt{16 \times 3} \times \sqrt{-1}=\pm 4\sqrt{3} i \\ \\
x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}=\frac{-(-4) \pm 4\sqrt{3}i}{2 \times 16}=\frac{4 \pm 4\sqrt{3}i}{32}=\frac{4(1 \pm \sqrt{3}i)}{4 \times 8}=\frac{1 \pm \sqrt{3}i}{8} \\
x=\frac{1-\sqrt{3}i}{8} \ \lor \ x=\frac{1+\sqrt{3}i}{8}[/tex]
The final answer:
[tex]\boxed{x=-\frac{1}{4} \hbox{ or } x=\frac{1-\sqrt{3}i}{8} \hbox{ or } x=\frac{1+\sqrt{3}i}{8}}[/tex]
