4e^((7pi)/(4)i)i+2e^((3pi)/(2)i)

Simplify e^(7pi/4)i: Step 1: Simplify e^((7pi)/(4)i) Using Euler's formula, e^(iθ) = cos(θ) + i*sin(θ), e^((7pi)/(4)i) = cos(7pi/4) + i*sin(7pi/4) = cos(-pi/4) + i*sin(-pi/4) // since cos and sin are periodic with period 2π = sqrt(2)/2 - i*sqrt(2)/2Calculate 4e^(7pi/4)i: Step 2: Calculate 4e^((7pi)/(4)i)i Multiply the result from Step 1 by 4, 4(sqrt(2)/2 - i*sqrt(2)/2) = 2sqrt(2) - 2i*sqrt(2) Now, multiply by i, (2sqrt(2) - 2i*sqrt(2))i = -2sqrt(2)i - 2sqrt(2)Simplify e^(3pi/2)i: Step 3: Simplify e^((3pi)/(2)i) Using Euler's formula again, e^((3pi)/(2)i) = cos(3pi/2) + i*sin(3pi/2) = 0 - i = -iCalculate 2e^(3pi/2)i: Step 4: Calculate 2e^((3pi)/(2)i) Multiply the result from Step 3 by 2, 2(-i) = -2iAdd results from Step 2 and Step 4: Step 5: Add the results from Step 2 and Step 4 (-2sqrt(2)i - 2sqrt(2)) + (-2i) = -2sqrt(2) - 4i*sqrt(2)

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$4 e^{\frac{7 \pi}{4} i} i+2 e^{\frac{3 \pi}{2} i}$

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4 e^{\frac{7 \pi}{4} i} i+2 e^{\frac{3 \pi}{2} i}

Simplify $e^{\frac{7\pi}{4}i}$ : Step $1$ : Simplify $e^{\left(\frac{7\pi}{4}\right)i}$ Using Euler's formula, $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ , $e^{\left(\frac{7\pi}{4}\right)i} = \cos\left(\frac{7\pi}{4}\right) + i\sin\left(\frac{7\pi}{4}\right) = \cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right) // since \cos and \sin are periodic with period 2\pi = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}$
Calculate $4e^{(7\pi/4)i}$ : Step $2$ : Calculate $4e^{((7\pi)/(4)i)i}$ $\newline$ Multiply the result from Step $1$ by $4$ , $\newline$ $4(\sqrt{2}/2 - i\sqrt{2}/2) = 2\sqrt{2} - 2i\sqrt{2}$ $\newline$ Now, multiply by $i$ , $\newline$ $(2\sqrt{2} - 2i\sqrt{2})i = -2\sqrt{2}i - 2\sqrt{2}$
Simplify $e^{\frac{3\pi}{2}i}$ : Step $3$ : Simplify $e^{\left(\frac{3\pi}{2}\right)i}$ $\newline$ Using Euler's formula again, $\newline$ $e^{\left(\frac{3\pi}{2}\right)i} = \cos\left(\frac{3\pi}{2}\right) + i\sin\left(\frac{3\pi}{2}\right)$ $\newline$ $= 0 - i$ $\newline$ $= -i$
Calculate $2e^{(3\pi/2)i}$ : Step $4$ : Calculate $2e^{\left(\frac{3\pi}{2}\right)i}$ $\newline$ Multiply the result from Step $3$ by $2$ , $\newline$ $2(-i) = -2i$
Add results from Step $2$ and Step $4$ : Step $5$ : Add the results from Step $2$ and Step $4$ $\newline$ $(-2\sqrt{2}i - 2\sqrt{2}) + (-2i)$ $\newline$ = $-2\sqrt{2} - 4i\sqrt{2}$