Evaluate the following square root expression.

(−i)10−36⎯⎯⎯⎯⎯⎯⎯√

To evaluate the given expression:

\[
(-i)^{10} - \sqrt{36}
\]

We can break this down into two parts: \( (-i)^{10} \) and \( \sqrt{36} \).

### Step 1: Evaluate \( (-i)^{10} \)
The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), and powers of \( i \) cycle in a repeating pattern every 4 terms:

\[
i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1
\]

Now, let's evaluate \( (-i)^{10} \). We can express this as \( (-i)^2 \) raised to the power of 5:

\[
(-i)^{10} = \left( (-i)^2 \right)^5 = (-1)^5 = -1
\]

### Step 2: Evaluate \( \sqrt{36} \)
The square root of 36 is:

\[
\sqrt{36} = 6
\]

### Step 3: Subtract the results
Now, subtract the results from Step 1 and Step 2:

\[
(-i)^{10} - \sqrt{36} = -1 - 6 = -7
\]

Thus, the value of the expression is:

\[
\boxed{-7}
\]

Would you like more details or further clarification?

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Here are 5 related questions to consider:
1. What is the general formula for powers of \( i \)?
2. How do you simplify powers of negative imaginary numbers like \( (-i) \)?
3. Can you evaluate \( (-i)^{15} \) using a similar approach?
4. How do complex numbers relate to Euler's formula?
5. What are the square roots of negative numbers in terms of \( i \)?

**Tip:** When working with powers of \( i \), remember the cyclical nature of the powers: \( i, -1, -i, 1 \). This can help simplify complex expressions.

Evaluate the following square root expression: (−i)^10−√36

Step-by-step guide on how to evaluate the complex expression (−i)^10−√36. This problem explores the cyclical nature of powers of the imaginary unit i and basic square root operations. Suitable for Grades 10-12.

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