The value of `[sqrt(2)(cos(56^(@)15^('))+isin(56^(@)15^('))]^(8)`, is

Options

A. `4i`
B. `8i`
C. `16i`
D. `-16i`

Answer

To solve the problem, we need to find the value of \([ \sqrt{2} (\cos(56^\circ 15') + i \sin(56^\circ 15')]^{8}\).

### Step-by-Step Solution:

1. **Convert Minutes to Degrees**:
- We know that \(1 \text{ minute} = \frac{1}{60} \text{ degrees}\).
- Therefore, \(15' = 15 \times \frac{1}{60} = \frac{1}{4} \text{ degrees}\).
- Thus, \(56^\circ 15' = 56^\circ + \frac{1}{4}^\circ = 56.25^\circ\).

2. **Express in Polar Form**:
- We can express the complex number in polar form as:
\[ z = \sqrt{2} \left( \cos(56.25^\circ) + i \sin(56.25^\circ) \right) \]
- This can be rewritten using Euler's formula:
\[ z = \sqrt{2} e^{i \cdot 56.25^\circ} \]

3. **Raise to the Power of 8**:
- We need to raise \(z\) to the power of 8:
\[ z^8 = \left( \sqrt{2} e^{i \cdot 56.25^\circ} \right)^8 \]
- This simplifies to:
\[ z^8 = (\sqrt{2})^8 \cdot (e^{i \cdot 56.25^\circ})^8 \]
- Calculating \((\sqrt{2})^8\):
\[ (\sqrt{2})^8 = 2^{4} = 16 \]
- And for the exponential part:
\[ (e^{i \cdot 56.25^\circ})^8 = e^{i \cdot (8 \cdot 56.25^\circ)} = e^{i \cdot 450^\circ} \]

4. **Simplify the Angle**:
- Since \(450^\circ\) can be simplified by subtracting \(360^\circ\):
\[ 450^\circ - 360^\circ = 90^\circ \]
- Thus, \(e^{i \cdot 450^\circ} = e^{i \cdot 90^\circ}\).

5. **Express in Rectangular Form**:
- Now, using Euler's formula again:
\[ e^{i \cdot 90^\circ} = \cos(90^\circ) + i \sin(90^\circ) = 0 + i \cdot 1 = i \]

6. **Final Result**:
- Therefore, we have:
\[ z^8 = 16 \cdot i \]

Thus, the value of \([ \sqrt{2} (\cos(56^\circ 15') + i \sin(56^\circ 15')]^{8}\) is \(16i\).