Write the expression in rectangular form, , and in exponential form,

Ask by Martin Dickson. in the United States

Dec 15,2024

Question

Write the expression in rectangular form, $$ x+y i $$, and in exponential form, $$ \mathrm{r} e^{i 	heta} $$ $$ (-1-i)^{3} $$

UpStudy AI · Accepted Answer

To express $$(-1 - i)^3$$ in both rectangular form $$x + yi$$ and exponential form $$re^{i\theta}$$, we will follow these steps:

### Step 1: Convert to Polar Form

First, we need to convert $$-1 - i$$ into polar form.

1. **Magnitude (r)**:
   $$
   r = | -1 - i | = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
   $$

2. **Argument ($$\theta$$)**:
   The argument $$\theta$$ can be found using:
   $$
   \theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-1}{-1}\right) = \tan^{-1}(1)
   $$
   Since $$-1 - i$$ is in the third quadrant, we add $$\pi$$:
   $$
   \theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}
   $$

Thus, we can express $$-1 - i$$ in polar form as:
$$
-1 - i = \sqrt{2} e^{i \frac{5\pi}{4}}
$$

### Step 2: Raise to the Power of 3

Now we raise this expression to the power of 3:
$$
(-1 - i)^3 = \left(\sqrt{2} e^{i \frac{5\pi}{4}}\right)^3
$$
Using the properties of exponents:
$$
= (\sqrt{2})^3 e^{i \frac{5\pi}{4} \cdot 3} = 2^{3/2} e^{i \frac{15\pi}{4}}
$$
Calculating $$2^{3/2}$$:
$$
2^{3/2} = 2 \sqrt{2}
$$

Now, simplify the argument:
$$
\frac{15\pi}{4} = 3\pi + \frac{3\pi}{4} = 3\pi + \frac{3\pi}{4} - 2\pi = \pi + \frac{3\pi}{4} = \frac{7\pi}{4}
$$

Thus, we have:
$$
(-1 - i)^3 = 2\sqrt{2} e^{i \frac{7\pi}{4}}
$$

### Step 3: Convert Back to Rectangular Form

Now we convert $$2\sqrt{2} e^{i \frac{7\pi}{4}}$$ back to rectangular form:
$$
e^{i \frac{7\pi}{4}} = \cos\left(\frac{7\pi}{4}\right) + i \sin\left(\frac{7\pi}{4}\right)
$$
Calculating the cosine and sine:
$$
\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}
$$
Thus:
$$
e^{i \frac{7\pi}{4}} = \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}
$$
Now multiply by $$2\sqrt{2}$$:
$$
2\sqrt{2} \left(\frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}\right) = 2 - 2i
$$

### Final Results

In rectangular form:
$$
(-1 - i)^3 = 2 - 2i
$$

In exponential form:
$$
(-1 - i)^3 = 2\sqrt{2} e^{i \frac{7\pi}{4}}
$$

Thus, the final answers are:
- Rectangular form: $$2 - 2i$$
- Exponential form: $$2\sqrt{2} e^{i \frac{7\pi}{4}}$$
Rectangular form: $$2 - 2i$$ Exponential form: $$2\sqrt{2} e^{i \frac{7\pi}{4}}$$

Write the expression in rectangular form, , and in exponential form,

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