write the square root of negative54 in its simpliewt form

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22-01-2025

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The square root of -54, written as √-54, presents a unique challenge because we're dealing with a negative number under the square root symbol. Let's break down how to simplify this expression into its simplest form.

Understanding Imaginary Numbers

The key to solving this lies in understanding imaginary numbers. Since no real number multiplied by itself equals a negative number, mathematicians defined the imaginary unit, i, as:

i = √-1

This allows us to work with the square roots of negative numbers.

Simplifying √-54 Step-by-Step

1. Factor out -1: We can rewrite √-54 as √(-1 * 54).

2. Separate the terms: Using the property of square roots (√(ab) = √a * √b), we can separate this into √-1 * √54.

3. Substitute i: Since √-1 = i, we can replace √-1 with i, giving us i√54.

4. Simplify the radical: Now we need to simplify √54. Let's find its prime factorization: 54 = 2 * 3 * 3 * 3 = 2 * 3³.

5. Extract perfect squares: We can rewrite √54 as √(9 * 6) = √9 * √6 = 3√6.

6. Combine terms: Substituting this back into our expression, we get 3√6 * i or, more conventionally written as 3i√6.

Therefore, the simplest form of √-54 is 3i√6.

Important Considerations:

• Order of operations: Remember the order of operations (PEMDAS/BODMAS). Deal with the negative sign under the radical first by factoring out -1 and introducing i.

• Imaginary vs. Complex Numbers: 3i√6 is an imaginary number. If you had a real number component in addition to the imaginary part (e.g., 2 + 3i√6), then you would have a complex number.

• Practice: Practicing with similar problems will help solidify your understanding of simplifying radicals involving imaginary numbers. Try simplifying √-72 or √-108 to further your skills.

This explanation provides a clear and concise way to simplify the square root of negative 54, explaining each step in detail. Remember to always factor out -1 and introduce the imaginary unit, i, when dealing with the square root of a negative number.