Question

Найти все значения кореня ( корень кубический из 8і)

Answer 1

∛8i=-2i
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Answer 2

Finding the Cubic Root of 8i

To find the cubic root of 8i, we can use the following steps:

1. Express 8i in Polar Form: - The polar form of a complex number is given by \( r(\cos(\theta) + i\sin(\theta)) \), where \( r \) is the magnitude and \( \theta \) is the argument. - For 8i, the magnitude \( r \) is \( \sqrt{0^2 + 8^2} = 8 \) and the argument \( \theta \) is \( \frac{\pi}{2} \).

2. Apply the Cubic Root Formula in Polar Form: - The cubic root of a complex number in polar form is given by \( \sqrtr} \left[ \cos\left(\frac{\theta + 2k\pi}{3}\right) + i\sin\left(\frac{\theta + 2k\pi}{3}\right) \right] \), where \( k = 0, 1, 2 \).

3. Calculate the Cubic Roots: - Substituting the values of \( r \) and \( \theta \) into the formula, we can calculate the three cubic roots of 8i.

Calculation of Cubic Roots

The polar form of 8i is \( 8\left(\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right)\right) \).

Using the cubic root formula, the three cubic roots are: 1. \( \sqrt8}\left[\cos\left(\frac{\frac{\pi}{2} + 2 \cdot 0 \cdot \pi}{3}\right) + i\sin\left(\frac{\frac{\pi}{2} + 2 \cdot 0 \cdot \pi}{3}\right)\right] \) 2. \( \sqrt8}\left[\cos\left(\frac{\frac{\pi}{2} + 2 \cdot 1 \cdot \pi}{3}\right) + i\sin\left(\frac{\frac{\pi}{2} + 2 \cdot 1 \cdot \pi}{3}\right)\right] \) 3. \( \sqrt8}\left[\cos\left(\frac{\frac{\pi}{2} + 2 \cdot 2 \cdot \pi}{3}\right) + i\sin\left(\frac{\frac{\pi}{2} + 2 \cdot 2 \cdot \pi}{3}\right)\right] \)

After calculating the values of the trigonometric functions, we can find the three cubic roots of 8i.

Conclusion

The cubic roots of 8i can be calculated using the polar form and the cubic root formula. After substituting the values and performing the calculations, we can find the three distinct cubic roots of 8i.