To convert the given complex numbers $z = 2e^{i\frac{3\pi}{4}}$ and $w = 2e^{-i\frac{\pi}{2}}$ into the form $x + yi$, and calculate $z + w$, follow these steps:

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Step 1: Convert $z = 2e^{i\frac{3\pi}{4}}$ to Cartesian form

The exponential form of a complex number $re^{i\theta}$ corresponds to the Cartesian form: $x + yi = r(\cos\theta + i\sin\theta).$ Here:

• $r = 2$,
• $\theta = \frac{3\pi}{4}$.

Using trigonometric values: $\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}, \quad \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}.$

Thus: $z = 2\left(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right) = -\sqrt{2} + i\sqrt{2}.$

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Step 2: Convert $w = 2e^{-i\frac{\pi}{2}}$ to Cartesian form

Similarly:

• $r = 2$,
• $\theta = -\frac{\pi}{2}$.

Using trigonometric values: $\cos\left(-\frac{\pi}{2}\right) = 0, \quad \sin\left(-\frac{\pi}{2}\right) = -1.$

Thus: $w = 2\left(0 - i\right) = -2i.$

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Step 3: Calculate $z + w$

Now add the Cartesian forms of $z$ and $w$: $z = -\sqrt{2} + i\sqrt{2}, \quad w = -2i.$

Add the real and imaginary parts separately: $z + w = (-\sqrt{2}) + (i\sqrt{2} - 2i).$

Simplify: $z + w = -\sqrt{2} - i(2 - \sqrt{2}).$

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Final Answer:

The result in Cartesian form is: $z + w = -\sqrt{2} - i(2 - \sqrt{2}).$

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Would you like more details on any step? Here are some related questions to explore further:

1. How do Euler's formula and trigonometric identities relate to complex number conversions?
2. How would you find the magnitude and angle of a complex number given in Cartesian form?
3. How do operations like addition and multiplication of complex numbers behave in polar form?
4. Can the result $z + w$ be converted back into polar form?
5. What is the geometric interpretation of adding $z$ and $w$ in the complex plane?

Tip: For complex number calculations, keep a list of common trigonometric values for standard angles handy—it simplifies conversions!