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  1. Beranda
  2. Kalkulator
  3. Kalkulator: Aljabar II
  4. Kalkulator Pra-Kalkulus

Tentukan $$$\sqrt{32 + 4 \sqrt{17} i}$$$

Kalkulator ini akan menemukan semua akar ke-$$$n$$$ ($$$n = 2$$$) dari bilangan kompleks $$$32 + 4 \sqrt{17} i$$$, dengan langkah-langkah yang ditampilkan.

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Masukan Anda

Temukan $$$\sqrt{32 + 4 \sqrt{17} i}$$$.

Solusi

Bentuk polar dari $$$32 + 4 \sqrt{17} i$$$ adalah $$$36 \left(\cos{\left(\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} \right)}\right)$$$ (untuk langkah-langkah, lihat kalkulator bentuk polar).

Menurut Rumus De Moivre, semua akar ke-$$$n$$$ dari bilangan kompleks $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$ diberikan oleh $$$r^{\frac{1}{n}} \left(\cos{\left(\frac{\theta + 2 \pi k}{n} \right)} + i \sin{\left(\frac{\theta + 2 \pi k}{n} \right)}\right)$$$, $$$k=\overline{0..n-1}$$$.

Diketahui bahwa $$$r = 36$$$, $$$\theta = \operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}$$$, dan $$$n = 2$$$.

  • $$$k = 0$$$: $$$\sqrt{36} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} + 2\cdot \pi\cdot 0}{2} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} + 2\cdot \pi\cdot 0}{2} \right)}\right) = 6 \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)}\right) = 6 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)} + 6 i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)}$$$
  • $$$k = 1$$$: $$$\sqrt{36} \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} + 2\cdot \pi\cdot 1}{2} \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} + 2\cdot \pi\cdot 1}{2} \right)}\right) = 6 \left(\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} + \pi \right)} + i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} + \pi \right)}\right) = - 6 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)} - 6 i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)}$$$

Jawaban

$$$\sqrt{32 + 4 \sqrt{17} i} = 6 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)} + 6 i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)}\approx 5.8309518948453 + 1.414213562373095 i$$$A

$$$\sqrt{32 + 4 \sqrt{17} i} = - 6 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)} - 6 i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{2} \right)}\approx -5.8309518948453 - 1.414213562373095 i$$$A


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