(1+x)^20=2.844

asked by guest
on Nov 25, 2024 at 10:53 pm



You asked:

Solve the equation \({\left( 1 + x \right)}^{20} = 2.844\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \left(-1 + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} \cos{\left(\frac{\pi k}{10} \right)}}{10}\right) + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} \sin{\left(\frac{\pi k}{10} \right)}}{10} i\] 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: \[\begin{aligned}x &= -1 + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000}}{10} \\&\approx 0.053650262\\x &= -1 - \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000}}{10} \\&\approx -2.0536503\end{aligned}\]

Solutions with nonzero imaginary part (\(8\) of \(18\) displayed): \[\begin{aligned}x &= -1 + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{10} + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} i \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{10} \\&\approx 0.0020809478 + 0.32559584 i\\x &= -1 + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} \cdot \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{10} + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{10} \\&\approx -0.14757903 + 0.61932009 i\\x &= -1 + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{10} + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} i \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{10} \\&\approx -0.38067991 + 0.85242097 i\\x &= -1 + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{10} + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{10} \\&\approx -0.67440416 + 1.0020809 i\\x &= -1 + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} i}{10} \\&= -1 + 1.0536503 i\\x &= -1 + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} \cdot \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{10} + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{10} \\&\approx -1.3255958 + 1.0020809 i\\x &= -1 - \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{10} + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} i \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{10} \\&\approx -1.6193201 + 0.85242097 i\\x &= -1 + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right)}{10} + \frac{\sqrt[10]{3} \cdot \sqrt[20]{31600000000000000000} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{10} \\&\approx -1.852421 + 0.61932009 i\end{aligned}\]

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

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