Express each of the following in the form b or bi, where b is a real number
`sqrt((-1)/(3))`

Answer

To express the given expression \(\sqrt{\frac{-1}{3}}\) in the form \(b\) or \(bi\), where \(b\) is a real number, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ z = \sqrt{\frac{-1}{3}} \]
Since we have a negative number inside the square root, we can express it using the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus, we can rewrite the expression as:
\[ z = \sqrt{\frac{-1}{3}} = \sqrt{-1} \cdot \sqrt{\frac{1}{3}} = i \cdot \sqrt{\frac{1}{3}} \]

### Step 2: Simplify the square root
Next, we simplify \(\sqrt{\frac{1}{3}}\):
\[ \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]
So, we can substitute this back into our expression for \(z\):
\[ z = i \cdot \frac{1}{\sqrt{3}} = \frac{i}{\sqrt{3}} \]

### Step 3: Identify \(b\)
Now, we can express \(z\) in the form \(bi\):
\[ z = 0 + \frac{1}{\sqrt{3}} i \]
Here, we can see that \(b = \frac{1}{\sqrt{3}}\).

### Final Result
Thus, the expression \(\sqrt{\frac{-1}{3}}\) can be expressed in the form \(bi\) as:
\[ z = \frac{i}{\sqrt{3}} \]
where \(b = \frac{1}{\sqrt{3}}\).

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