Questions: Evaluate the following square root expression. (-i)^10 sqrt-169

Solution Steps

To evaluate the given expression, we need to break it down into two parts: \((-i)^{10}\) and \(\sqrt{-169}\). First, we simplify \((-i)^{10}\) using properties of imaginary numbers. Then, we find the square root of \(-169\) by expressing it in terms of imaginary numbers.

To evaluate \((-i)^{10}\), we can express \(-i\) as \(-1 \cdot i\). Using the property of powers of imaginary numbers, we find: \[ (-i)^{10} = (-1)^{10} \cdot i^{10} = 1 \cdot (i^2)^5 = 1 \cdot (-1)^5 = -1 \] Thus, we have: \[ (-i)^{10} = -1 \]

Next, we calculate \(\sqrt{-169}\). This can be expressed as: \[ \sqrt{-169} = \sqrt{169} \cdot \sqrt{-1} = 13i \]

Now, we combine the results from Step 1 and Step 2: \[ (-i)^{10} \cdot \sqrt{-169} = -1 \cdot 13i = -13i \]

Final Answer