(x-1)^6= 64
You asked:
Solve the equation \({\left( x - 1 \right)}^{6} = 64\) for the variable \(x\).
MathBot Answer:
The solutions are given by the formula: \[x = \left(1 + 2 \cos{\left(\frac{\pi k}{3} \right)}\right) + 2 \sin{\left(\frac{\pi k}{3} \right)} i\] where \(k\) is an integer in \(0 \le k < 6\).
There are \(2\) real solutions.
There are \(4\) solutions with nonzero imaginary part.
Real solutions: \[\begin{aligned}x &= 3\\x &= -1\end{aligned}\]
Solutions with nonzero imaginary part: \[\begin{aligned}x &= 2 + \sqrt{3} i \\&= 2 + 1.7320508 i\\x &= \sqrt{3} i \\&\approx 1.7320508 i\\x &= - \sqrt{3} i \\&\approx -1.7320508 i\\x &= 2 - \sqrt{3} i \\&= 2 -1.7320508 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).