Questions: Finding a Power of a Complex Number use DeMoivre's Theorem to find the power of the complex number. Write the result in standard form. 26. (2+5i)^6

Solution Steps

To convert the complex number \( z = 2 + 5i \) to polar form, we calculate the modulus \( r \) and the argument \( \theta \): \[ r = |z| = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \] \[ \theta = \tan^{-1}\left(\frac{5}{2}\right) \]

Using DeMoivre's Theorem, we find \( z^6 \): \[ z^6 = r^6 \left( \cos(6\theta) + i \sin(6\theta) \right) \] Calculating \( r^6 \): \[ r^6 = \left(\sqrt{29}\right)^6 = 29^3 = 24389 \] Calculating \( 6\theta \): \[ 6\theta = 6 \cdot \tan^{-1}\left(\frac{5}{2}\right) \approx 7.14174 \]

Now we convert back to standard form: \[ z^6 = 24389 \left( \cos(7.14174) + i \sin(7.14174) \right) \] Calculating the cosine and sine values: \[ \cos(7.14174) \approx \frac{15939}{24389}, \quad \sin(7.14174) \approx \frac{18460}{24389} \] Thus, the result in standard form is: \[ z^6 \approx 15939 + 18460i \]

Final Answer