Square Root of -27

The square root of a number is a value that, when multiplied by itself, gives the original number. For negative numbers, like -27, the square root involves complex numbers since there is no real number whose square is negative. The square root of -27 can be expressed in terms of imaginary numbers as √(-27) = √(27) × i, where i is the imaginary unit defined as √(-1). In its simplified form, √(27) = 3√3, therefore √(-27) = 3√3i.

There are several methods to find the square root of a number, but for negative numbers, we transition to complex numbers. Here, we will explore:

1. Understanding imaginary numbers

2. Simplifying the square root of the positive part

3. Combining with the imaginary unit

Imaginary numbers are used to represent the square roots of negative numbers. The imaginary unit i is defined as √(-1). Therefore, the square root of any negative number can be rewritten using i. For example, √(-27) can be expressed as √(27) × i.

To simplify √(-27), first simplify √(27). The prime factorization of 27 is 3 × 3 × 3, which can be grouped as (3 × 3) × 3. This simplifies as:

Step 1: √(27) = √(3^2 × 3) = 3√3

Step 2: Hence, the square root of -27 is 3√3i

After simplifying the positive part, we include the imaginary unit to express the square root of -27:

Step 1: From √(27) = 3√3, we multiply by i to account for the negative sign.

Step 2: Therefore, the square root of -27 is 3√3i.

• Imaginary Number: A number that can be written as a real number multiplied by the imaginary unit i, where i = √(-1).
   
• Complex Number: A number comprising a real part and an imaginary part, expressed as a + bi.
   
• Conjugate: For a complex number a + bi, its conjugate is a - bi.
   
• Absolute Value (Magnitude): The distance of a complex number from the origin in the complex plane, calculated as √(a^2 + b^2) for a + bi.
   
• Square: The result of multiplying a number by itself, which for complex numbers involves both real and imaginary components.