Given that \( \mathrm{b}=2 e^{\frac{\pi i}{4}} \), what is \( b^{3} \) Select one: a. \( 8 e^{\frac{\pi i}{4}} \) b. \( 2 e^{\frac{3 \pi i}{4}} \) c. \( 6 e^{\frac{3 \pi i}{4}} \) d. \( 8 e^{\frac{3 \pi i}{4}} \)

To find \( b^{3} \) given that \( \mathrm{b}=2 e^{\frac{\pi i}{4}} \), we can use the properties of exponentiation. 1. First, we compute \( b^{3} = (2 e^{\frac{\pi i}{4}})^{3} \). 2. This can be rewritten as \( 2^{3} (e^{\frac{\pi i}{4}})^{3} \). 3. Calculating \( 2^{3} \) gives us \( 8 \). 4. For the exponential part, we have \( (e^{\frac{\pi i}{4}})^{3} = e^{\frac{3\pi i}{4}} \). Combining these results together, we get: \[ b^{3} = 8 e^{\frac{3\pi i}{4}} \] Thus, the answer is \( \boxed{8 e^{\frac{3 \pi i}{4}}} \), which corresponds to option d.