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The problem asks to express the radical \(\pm \sqrt{-77}\) using the imaginary unit \(i\) and simplify the expression.

### Solution:

1. The square root of a negative number involves the imaginary unit \(i\), where \(i = \sqrt{-1}\).
2. First, separate the negative sign inside the square root:
   \[
   \sqrt{-77} = \sqrt{-1 \times 77} = \sqrt{-1} \times \sqrt{77} = i \sqrt{77}
   \]
3. Therefore, \(\pm \sqrt{-77} = \pm i \sqrt{77}\).

The simplified form is:
\[
\boxed{\pm i \sqrt{77}}
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would you express \(\sqrt{-100}\) using the imaginary unit \(i\)?
2. What is the square root of a negative number conceptually in the complex plane?
3. How do you simplify the expression \(\sqrt{-x}\) for a general negative number \(x\)?
4. Can you use the imaginary unit \(i\) in addition and multiplication of complex numbers?
5. What are the properties of the imaginary unit \(i\) in powers, such as \(i^2\), \(i^3\), etc.?

**Tip:** Always remember that \(\sqrt{-1} = i\), and when working with radicals of negative numbers, factor out the \(i\) first before simplifying further.

Express the radical using the imaginary unit, i. ±√-77

Learn how to express the square root of -77 using the imaginary unit i, with a detailed step-by-step solution. This problem covers key concepts from complex numbers and radicals, making it ideal for students in Grades 9-12.

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