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Chapter 6: Problem 61

Use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$(1+i)^{5}$$

Short Answer

Expert verified
The fifth power of the complex number \(1+i\) in rectangular form is \(0 - 8i.\)

Step by step solution

01

Express the Complex Number in Trigonometric Form

Rewrite the complex number \(1+i\) into trigonometric form, \(r(\cos \theta + i \sin \theta)\). To do this, we need to find the modulus \(r = \sqrt{1^2 + 1^2} = \sqrt{2}\), and the argument \(\theta = \tan^{-1}(1/1) = \tan^{-1}(1) = \pi/4\). Therefore, \(1+i\) can be written as \(r(\cos \theta + i \sin \theta) = \sqrt{2}(\cos(\pi/4) + i\sin(\pi/4))\).
02

Apply De Moivre's Theorem

Using DeMoivre's theorem, we can raise the complex number to the desired power. According to DeMoivre's theorem, \((r(\cos \theta + i \sin \theta))^n = r^n (\cos n\theta + i \sin n\theta)\). In this case, \(n\) equals 5. So applying the theorem gives \((\sqrt{2}(\cos(\pi/4) + i\sin(\pi/4)))^5 = (\sqrt{2})^5 ((\cos(5*\pi/4) + i\sin(5*\pi/4))\). Simplicity this expression and we get \(2^{5/2}(\cos(5\pi/4) + i\sin(5\pi/4))\). Using the unit circle, we can find that \(\cos(5\pi/4) = -1/\sqrt{2}\) and \(\sin(5\pi/4) = -1/\sqrt{2}\). Thus the expression simplifies to \(2^{5/2}(-1/\sqrt{2} - i/\sqrt{2})\), therefore we get -8 \(i\).
03

Convert back to Rectangular Form

Finally, converting the result back to rectangular form will just be to express the complex number as \(a + bi\) where \(a\) is the real part and \(b\) is the imaginary part. Our final answer then is \(0 - 8i.\)

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