27000/5000= (1+r/4)^24

asked by guest
on Sep 27, 2024 at 10:22 am



You asked:

Solve the equation \(\frac{27000}{5000} = {\left( 1 + \frac{r}{4} \right)}^{24}\) for the variable \(r\).

MathBot Answer:

The solutions are given by the formula: \[r = \left(-4 + \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} \cos{\left(\frac{\pi k}{12} \right)}}{5}\right) + \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} \sin{\left(\frac{\pi k}{12} \right)}}{5} i\] where \(k\) is an integer in \(0 \le k < 24\).

There are \(2\) real solutions.

There are \(22\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}r &= -4 + \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}}}{5} \\&\approx 0.2911767\\r &= -4 - \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}}}{5} \\&\approx -8.2911767\end{aligned}\]

Solutions with nonzero imaginary part (\(8\) of \(22\) displayed): \[\begin{aligned}r &= -4 + \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{5} + \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} i \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{5} \\&\approx 0.1449584 + 1.1106383 i\\r &= -4 + \frac{2 \cdot 3^{\frac{5}{8}} \cdot 5^{\frac{23}{24}}}{5} + \frac{2 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} i}{5} \\&\approx -0.28373197 + 2.1455883 i\\r &= -4 + \frac{2 \sqrt{2} \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}}}{5} + \frac{2 \sqrt{2} \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} i}{5} \\&\approx -0.96567986 + 3.0343201 i\\r &= -4 + \frac{2 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}}}{5} + \frac{2 \cdot 3^{\frac{5}{8}} \cdot 5^{\frac{23}{24}} i}{5} \\&\approx -1.8544117 + 3.716268 i\\r &= -4 + \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{5} + \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} i \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{5} \\&\approx -2.8893617 + 4.1449584 i\\r &= -4 + \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} i}{5} \\&= -4 + 4.2911767 i\\r &= -4 + \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} \left(- \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}\right)}{5} + \frac{4 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}} i \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{5} \\&\approx -5.1106383 + 4.1449584 i\\r &= -4 - \frac{2 \cdot \sqrt[8]{3} \cdot 5^{\frac{23}{24}}}{5} + \frac{2 \cdot 3^{\frac{5}{8}} \cdot 5^{\frac{23}{24}} i}{5} \\&\approx -6.1455884 + 3.716268 i\end{aligned}\]

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

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