Express in polar form (mod-Amplitude form)
`-sqrt7 + isqrt(21)`

Answer

To express the complex number \(-\sqrt{7} + i\sqrt{21}\) in polar form, we will follow these steps:

### Step 1: Identify the complex number
Let \( z = -\sqrt{7} + i\sqrt{21} \).

### Step 2: Calculate the modulus \( r \)
The modulus \( r \) of a complex number \( z = a + bi \) is given by:
\[ r = \sqrt{a^2 + b^2} \]
Here, \( a = -\sqrt{7} \) and \( b = \sqrt{21} \).

Calculating \( r \):
\[ r = \sqrt{(-\sqrt{7})^2 + (\sqrt{21})^2} \]
\[ = \sqrt{7 + 21} \]
\[ = \sqrt{28} \]
\[ = 2\sqrt{7} \]

### Step 3: Calculate the angle \( \theta \)
The angle \( \theta \) can be found using the formulas:
\[ \cos \theta = \frac{a}{r} \quad \text{and} \quad \sin \theta = \frac{b}{r} \]
Substituting the values:
\[ \cos \theta = \frac{-\sqrt{7}}{2\sqrt{7}} = -\frac{1}{2} \]
\[ \sin \theta = \frac{\sqrt{21}}{2\sqrt{7}} = \frac{\sqrt{3}}{2} \]

### Step 4: Determine the quadrant
Since \( \cos \theta \) is negative and \( \sin \theta \) is positive, \( \theta \) is in the second quadrant.

### Step 5: Find the angle \( \theta \)
From trigonometric values, we know:
\[ \cos 120^\circ = -\frac{1}{2} \quad \text{and} \quad \sin 120^\circ = \frac{\sqrt{3}}{2} \]
Thus, \( \theta = 120^\circ \).

### Step 6: Write the polar form
The polar form of the complex number is given by:
\[ z = r(\cos \theta + i\sin \theta) \]
Substituting the values of \( r \) and \( \theta \):
\[ z = 2\sqrt{7} \left( \cos 120^\circ + i \sin 120^\circ \right) \]

### Final Answer
Thus, the polar form of the complex number \(-\sqrt{7} + i\sqrt{21}\) is:
\[ z = 2\sqrt{7} \left( \cos 120^\circ + i \sin 120^\circ \right) \]

---