To solve for the expression \( z = \sqrt{5} - \sqrt{3i} \), we need to find the values for the square roots involved and understand the properties of complex numbers. Let's break it down step-by-step.
1. Square root of 5:
- The square root of 5 is a straightforward real number. \(\sqrt{5} \approx 2.236\).
2. Square root of \(3i\):
- To find the square root of a complex number, we need to convert \(3i\) into its polar form.
- Recall that \(3i = 0 + 3i\). The modulus \( r \) of \( 3i \) is \(|3i| = 3\).
- The argument \( \theta \) (also known as the angle) for \( 3i \) is \( \pi/2 \) because it lies on the positive imaginary axis.
- Thus, \( 3i \) in polar form is \( 3(\cos(\pi/2) + i\sin(\pi/2)) \).
3. Square root in polar form:
- The square root of a complex number \( re^{i\theta} \) is given by \(\sqrt{r} e^{i\theta/2} \).
- Therefore, \( \sqrt{3i} = \sqrt{3} e^{i \pi/4} = \sqrt{3} (\cos(\pi/4) + i\sin(\pi/4)) \).
- Substituting the values, we get \( \sqrt{3} (\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}) \).
- This simplifies to \( \sqrt{3} \cdot \frac{1}{\sqrt{2}} + i \sqrt{3} \cdot \frac{1}{\sqrt{2}} \).
- Which can be further simplified to \( \sqrt{\frac{3}{2}} + i \sqrt{\frac{3}{2}} \)
- \(\sqrt{\frac{3}{2}} \approx 1.224 \).
Therefore, \( \sqrt{3i} \approx 1.224 + 1.224i \).
4. Combining the results:
- Now, substitute these values back into the original expression \( z = \sqrt{5} - \sqrt{3i} \).
- We get \( z \approx 2.236 - (1.224 + 1.224i) \).
- Distribute the negative sign: \( z \approx 2.236 - 1.224 - 1.224i \).
- Combine the real parts: \( 2.236 - 1.224 = 1.012 \).
- So, \( z \approx 1.012 - 1.224i \).
Thus, the solution to the expression \( z = \sqrt{5} - \sqrt{3i} \) is approximately:
[tex]\[ z \approx 1.0113231061082 - 1.22474487139159i \][/tex]
