write the square root of -54 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 it into its simplest form.

Understanding Imaginary Numbers

The key to simplifying √-54 lies in understanding imaginary numbers. Since no real number, when multiplied by itself, results in a negative number, mathematicians defined the imaginary unit, i, as follows:

i² = -1

This means that i = √-1. This allows us to work with square roots of negative numbers.

Simplifying √-54 Step-by-Step

1. Factor out -1: The first step is to separate the negative sign from the number:

√-54 = √(-1 * 54)

2. Simplify √-1: We know that √-1 = i, so we can substitute:

√(-1 * 54) = i√54

3. Find Perfect Square Factors: Now let's focus on simplifying √54. We need to find the largest perfect square that divides evenly into 54. That number is 9 (because 9 x 6 = 54).

4. Separate the Perfect Square: Rewrite √54 using the perfect square we found:

i√54 = i√(9 * 6)

5. Simplify the Perfect Square: The square root of 9 is 3, so:

i√(9 * 6) = i * 3√6

6. Final Simplified Form: This gives us our final, simplified answer:

3i√6

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

Key Concepts to Remember

• Imaginary Unit (i): i = √-1, and i² = -1. This is fundamental to working with the square roots of negative numbers.
• Perfect Squares: Knowing your perfect squares (1, 4, 9, 16, 25, etc.) is crucial for simplifying radicals.
• Factoring: Breaking down the number under the square root into its prime factors helps you identify perfect squares.

This process allows us to express the square root of a negative number in a clear, concise, and mathematically accurate way using imaginary numbers. Remember, the solution is not a real number but an imaginary number, a crucial concept in higher-level mathematics.