Math Problem Statement

Consider the complex number z=5⋅√2+5⋅√2⋅i

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Use De Moivre's Theorem to find z4

.

z4=

Solution

Given the complex number z=52+52iz = 5\sqrt{2} + 5\sqrt{2}i, we need to find z4z^4 using De Moivre's Theorem.

Step 1: Convert to Polar Form

A complex number z=a+biz = a + bi can be represented in polar form as: z=r(cosθ+isinθ)z = r(\cos \theta + i \sin \theta) where:

  • r=a2+b2r = \sqrt{a^2 + b^2} is the modulus
  • θ=tan1(ba)\theta = \tan^{-1}\left(\frac{b}{a}\right) is the argument

For z=52+52iz = 5\sqrt{2} + 5\sqrt{2}i:

  • a=52a = 5\sqrt{2}
  • b=52b = 5\sqrt{2}

The modulus rr is calculated as: r=(52)2+(52)2=50+50=100=10r = \sqrt{(5\sqrt{2})^2 + (5\sqrt{2})^2} = \sqrt{50 + 50} = \sqrt{100} = 10

The argument θ\theta is: θ=tan1(5252)=tan1(1)=π4\theta = \tan^{-1}\left(\frac{5\sqrt{2}}{5\sqrt{2}}\right) = \tan^{-1}(1) = \frac{\pi}{4}

Thus, the polar form of zz is: z=10(cosπ4+isinπ4)z = 10 \left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right)

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that: zn=rn(cos(nθ)+isin(nθ))z^n = r^n \left(\cos(n\theta) + i\sin(n\theta)\right) For z4z^4: z4=104(cos(4×π4)+isin(4×π4))z^4 = 10^4 \left(\cos\left(4 \times \frac{\pi}{4}\right) + i \sin\left(4 \times \frac{\pi}{4}\right)\right)

Simplifying: z4=10000(cosπ+isinπ)z^4 = 10000 \left(\cos \pi + i \sin \pi\right)

Since cosπ=1\cos \pi = -1 and sinπ=0\sin \pi = 0, we have: z4=10000(1+0i)=10000z^4 = 10000(-1 + 0i) = -10000

Final Answer:

z4=10000z^4 = -10000

Would you like further details or explanations?

Here are some related questions you might find useful:

  1. How can we find the modulus of any complex number?
  2. What are the geometric interpretations of modulus and argument?
  3. How does De Moivre's Theorem apply to roots of complex numbers?
  4. What is the significance of the polar form of a complex number?
  5. Can De Moivre's Theorem be used for non-integer powers?

Tip: When dealing with complex numbers, always check the quadrant of the angle to correctly determine the argument.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Polar Form
De Moivre's Theorem

Formulas

Polar form of a complex number
De Moivre's Theorem

Theorems

De Moivre's Theorem

Suitable Grade Level

Advanced High School