write the square root of negative54 in its simpliewt form

Lussy04

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22-01-2025

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The square root of a negative number introduces the concept of imaginary numbers. Let's break down how to simplify √-54.

Understanding Imaginary Numbers

Before we begin, it's crucial to understand imaginary numbers. The square root of -1 is defined as i, where i² = -1. This allows us to work with the square roots of negative numbers.

Simplifying √-54

1. Factor out -1: The first step is to separate the negative sign from the number: √-54 = √(-1 * 54)

2. Separate the square root: We can rewrite this as: √(-1) * √54

3. Introduce i: Remember, √(-1) = i, so we now have: i√54

4. Simplify the square root of 54: Now we need to simplify √54. Find the largest perfect square that divides evenly into 54. That's 9 (since 9 x 6 = 54).

5. Rewrite and simplify: We can rewrite √54 as √(9 * 6). This simplifies to √9 * √6 = 3√6

6. Combine the terms: Putting it all together, we get: i * 3√6 which simplifies to 3i√6

Final Answer: The simplest form of √-54 is 3i√6

Key Concepts Revisited

• Imaginary Unit (i): The imaginary unit, i, is defined as the square root of -1. It's a fundamental concept in complex numbers.

• Perfect Squares: Recognizing perfect squares (like 9, 16, 25, etc.) is key to simplifying square roots.

• Complex Numbers: Numbers in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. Our simplified answer, 3i√6, is a complex number where 'a' = 0 and 'b' = 3√6.

This step-by-step approach helps you understand not just the answer but the underlying mathematical principles involved in simplifying square roots of negative numbers. Remember to always factor out the -1 and introduce the imaginary unit i when dealing with the square root of a negative number.