To confirm the correctness of the given statement, let's analyze it step by step:

Step 1: Polar Form of $\sqrt{-14i}$

The expression begins by rewriting $\sqrt{-14i}$ in polar form. Let's verify the polar representation.

1. Convert $-14i$ to polar coordinates:
   • The modulus is $|-14i| = \sqrt{14}$.
   • The argument (angle) is $-\pi/2$, as the number lies on the negative imaginary axis.

Thus: $\sqrt{-14i} = \sqrt{\sqrt{14}} \cdot \left(\cos\left(\frac{-\pi}{4}\right) + i\sin\left(\frac{-\pi}{4}\right)\right)$

This matches the expression in the first step.

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Step 2: Simplifying the Trigonometric Functions

For $\cos(-\pi/4)$ and $\sin(-\pi/4)$:

• $\cos(-\pi/4) = \frac{\sqrt{2}}{2}$
• $\sin(-\pi/4) = -\frac{\sqrt{2}}{2}$

Substituting these values, we get: $\sqrt{-14i} = \sqrt{\sqrt{14}} \cdot \left(\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\right)$

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Step 3: Factorizing Further

Now, factor out $\sqrt{14}$ and simplify: $\sqrt{-14i} = \sqrt{14} \cdot \frac{\sqrt{2}}{2} \cdot (1 - i)$

Combining terms: $\sqrt{-14i} = \sqrt{7} \cdot (1 - i)$

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Conclusion:

The statement provided is correct. It accurately expresses $\sqrt{-14i}$ in the simplified form as $\sqrt{7}(1 - i)$.

Let me know if you need further clarification or have additional questions. Here are five related questions for exploration:

1. How do you find the polar form of a complex number in general?
2. Why is the argument $-\pi/4$ used in this calculation?
3. What is the geometric meaning of $\sqrt{7}(1 - i)$?
4. How does the modulus affect the square root of complex numbers?
5. Can the square root of $-14i$ be expressed in another valid equivalent form?

Tip: When simplifying complex roots, always confirm both the modulus and argument carefully to avoid errors in sign or quadrant placement.