To transform the expression [tex]\(\left[\sqrt{7}\left(\cos \frac{3 \pi}{32} + i \sin \frac{3 \pi}{32}\right)\right]^8\)[/tex] into rectangular form and exponential form, let's go through the problem step-by-step.
### 1. Convert the given expression to exponential form
Firstly, rewrite [tex]\(\sqrt{7}\left(\cos \frac{3 \pi}{32} + i \sin \frac{3 \pi}{32}\right)\)[/tex] using Euler's formula, where [tex]\(e^{i\theta} = \cos \theta + i \sin \theta\)[/tex]:
[tex]\[
\sqrt{7}\left(\cos \frac{3 \pi}{32} + i \sin \frac{3 \pi}{32}\right) = \sqrt{7}e^{i \left(\frac{3 \pi}{32}\right)}
\][/tex]
### 2. Raise the expression to the power of 8
Using the properties of exponents, specifically De Moivre's Theorem [tex]\((r e^{i\theta})^n = r^n e^{i n \theta}\)[/tex], we get:
[tex]\[
\left[\sqrt{7} e^{i \left(\frac{3 \pi}{32}\right)}\right]^8 = \left(\sqrt{7}\right)^8 e^{i \left(8 \cdot \frac{3 \pi}{32}\right)}
\][/tex]
Now, calculate each component:
#### Compute [tex]\(\left(\sqrt{7}\right)^8\)[/tex]
[tex]\[
\left(\sqrt{7}\right)^8 = (7^{1/2})^8 = 7^{4} = 2401
\][/tex]
#### Compute the new angle [tex]\(\theta \cdot 8\)[/tex]
[tex]\[
8 \cdot \frac{3 \pi}{32} = \frac{24 \pi}{32} = \frac{3 \pi}{4}
\][/tex]
### 3. Exponential form
Substituting back into our exponential form expression:
[tex]\[
2401 e^{i \left(\frac{3 \pi}{4}\right)}
\][/tex]
### 4. Convert to rectangular form
Using Euler's formula, recall [tex]\(e^{i \theta} = \cos \theta + i \sin \theta\)[/tex]:
[tex]\[
2401 e^{i \left(\frac{3 \pi}{4}\right)} = 2401 \left(\cos \frac{3 \pi}{4} + i \sin \frac{3 \pi}{4}\right)
\][/tex]
Identify the trigonometric values:
[tex]\[
\cos \left(\frac{3 \pi}{4}\right) = -\frac{\sqrt{2}}{2} \quad \text{and} \quad \sin \left(\frac{3 \pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
Multiply them by 2401:
[tex]\[
2401 \left(-\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\right) = 2401 \left( -\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right)
\][/tex]
Simplify the expression:
[tex]\[
2401 \left( -\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right) = 2401 \left( -\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right) = -1697.763381628901 + 1697.763381628901 i
\][/tex]
### Summary
- The rectangular form of the given expression is [tex]\( -1697.763381628901 + 1697.763381628901 i \)[/tex].
- The exponential form of the given expression is [tex]\( 2401 e^{i \frac{3 \pi}{4}} \)[/tex].
