(400/1000) = (1/((1+x)^20))

asked by guest
on Nov 25, 2024 at 12:32 am



You asked:

Solve the equation \(\frac{400}{1000} = \frac{1}{{\left( 1 + x \right)}^{20}}\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \left(-1 + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} \cos{\left(\frac{\pi k}{10} \right)}}{2}\right) + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} \sin{\left(\frac{\pi k}{10} \right)}}{2} i\] where \(k\) is an integer in \(0 \le k < 20\).

There are \(2\) real solutions.

There are \(18\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= -1 + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5}}{2} \\&\approx 0.046880235\\x &= -1 - \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5}}{2} \\&\approx -2.0468802\end{aligned}\]

Solutions with nonzero imaginary part (\(8\) of \(18\) displayed): \[\begin{aligned}x &= -1 + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} i \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} \\&\approx -0.0043577307 + 0.32350378 i\\x &= -1 + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} \cdot \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} \\&\approx -0.1530561 + 0.61534076 i\\x &= -1 + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} i \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} \\&\approx -0.38465924 + 0.8469439 i\\x &= -1 + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} \\&\approx -0.67649622 + 0.99564227 i\\x &= -1 + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} i}{2} \\&= -1 + 1.0468802 i\\x &= -1 + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} \cdot \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2} + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} \\&\approx -1.3235038 + 0.99564227 i\\x &= -1 - \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} i \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} \\&\approx -1.6153408 + 0.8469439 i\\x &= -1 + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right)}{2} + \frac{2^{\frac{19}{20}} \cdot \sqrt[20]{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} \\&\approx -1.8469439 + 0.61534076 i\end{aligned}\]

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

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