d^5 =525316.46

asked by guest
on Jan 15, 2025 at 8:13 am



You asked:

Solve the equation \({d}^{5} = 525316.46\) for the variable \(d\).

MathBot Answer:

The solutions are given by the formula: \[d = \frac{\sqrt[5]{52531646000} \cos{\left(\frac{2 \pi k}{5} \right)}}{10} + \frac{\sqrt[5]{52531646000} \sin{\left(\frac{2 \pi k}{5} \right)}}{10} i\] where \(k\) is an integer in \(0 \le k < 5\).

There is \(1\) real solution.

There are \(4\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}d &= \frac{\sqrt[5]{52531646000}}{10} \\&\approx 13.934269\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}d &= \frac{\sqrt[5]{52531646000} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{10} + \frac{\sqrt[5]{52531646000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{10} \\&\approx 4.305926 + 13.252278 i\\d &= \frac{\sqrt[5]{52531646000} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right)}{10} + \frac{\sqrt[5]{52531646000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{10} \\&\approx -11.273061 + 8.190358 i\\d &= \frac{\sqrt[5]{52531646000} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right)}{10} - \frac{\sqrt[5]{52531646000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{10} \\&\approx -11.273061 -8.190358 i\\d &= \frac{\sqrt[5]{52531646000} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{10} - \frac{\sqrt[5]{52531646000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{10} \\&\approx 4.305926 -13.252278 i\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

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