To find the product of the complex numbers [tex]\( z \)[/tex] and [tex]\( w \)[/tex], we need to use the properties of complex numbers in polar form.
### Step-by-Step Solution
1. Identify the magnitudes and angles of [tex]\( z \)[/tex] and [tex]\( w \)[/tex]:
- For [tex]\( z \)[/tex]:
- Magnitude: [tex]\( |z| = 38 \)[/tex]
- Angle: [tex]\( \arg(z) = \frac{\pi}{8} \)[/tex]
- For [tex]\( w \)[/tex]:
- Magnitude: [tex]\( |w| = 2 \)[/tex]
- Angle: [tex]\( \arg(w) = \frac{\pi}{16} \)[/tex]
2. Calculate the magnitude of the product [tex]\( zw \)[/tex]:
- The magnitude of the product of two complex numbers is the product of their magnitudes:
[tex]\[
|zw| = |z| \cdot |w| = 38 \cdot 2 = 76
\][/tex]
3. Calculate the angle of the product [tex]\( zw \)[/tex]:
- The angle of the product of two complex numbers is the sum of their angles:
[tex]\[
\arg(zw) = \arg(z) + \arg(w) = \frac{\pi}{8} + \frac{\pi}{16}
\][/tex]
- To add these fractions, find a common denominator (which is 16):
[tex]\[
\frac{\pi}{8} = \frac{2\pi}{16}
\][/tex]
[tex]\[
\frac{2\pi}{16} + \frac{\pi}{16} = \frac{3\pi}{16}
\][/tex]
4. Express [tex]\( zw \)[/tex] in its final form:
[tex]\[
zw = 76 \left( \cos \left( \frac{3\pi}{16} \right) + i \sin \left( \frac{3\pi}{16} \right) \right)
\][/tex]
Therefore, the product [tex]\( zw \)[/tex] is:
[tex]\[
76 \left( \cos \left( \frac{3\pi}{16} \right) + i \sin \left( \frac{3\pi}{16} \right) \right)
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{76 \left( \cos \left( \frac{3\pi}{16} \right) + i \sin \left( \frac{3\pi}{16} \right) \right)}
\][/tex]
