Square Root of -28

The square root is the inverse of the square of a number. Since -28 is negative, its square root is not a real number. Instead, it is expressed in terms of imaginary numbers. The square root of -28 can be expressed as √(-28) = √(28) * √(-1) = 2√7 * i, where i is the imaginary unit such that i² = -1.

To find the square roots of negative numbers, we use imaginary numbers. The process involves separating the square root of the positive component and the imaginary unit. Let us explore the methods: Separation into real and imaginary components Prime factorization of the positive part Expressing in terms of i

First, we find the prime factorization of the positive component, 28.

Step 1: Prime factorization of 28 28 = 2 x 2 x 7 = 2² x 7

Step 2: Take the square root of the positive part separately √28 = √(2² x 7) = 2√7

Step 3: Combine with the imaginary unit Since the original number is negative, we multiply by the imaginary unit: 2√7 * i.

The square root of -28 involves the imaginary unit. Here's how to express it:

Step 1: Recognize the negative sign The negative sign indicates an imaginary component.

Step 2: Factor the positive and imaginary parts separately √(-28) = √(28) * √(-1)

Step 3: Solve for the components √28 = 2√7 √(-1) = i

Step 4: Express the result The result is 2√7 * i, representing the square root of -28.

Understanding square roots of negative numbers is crucial in fields involving complex analysis and electrical engineering. Imaginary numbers are used to model real-world phenomena like AC circuits and oscillations.

The complex plane is a tool used to visualize complex numbers. The square root of -28 can be represented as a point on this plane.

Step 1: The real part is 0

Step 2: The imaginary part is 2√7 This point (0, 2√7) is located on the imaginary axis, as there is no real component.

• Imaginary Unit: The imaginary unit i is defined such that i² = -1. It is used to express the square roots of negative numbers.
   
• Complex Number: A complex number has a real part and an imaginary part, expressed as a + bi, where a and b are real numbers.
   
• Prime Factorization: Breaking down a number into its prime components. For example, the prime factorization of 28 is 2² x 7.
   
• Modulus: The modulus of a complex number is its distance from the origin on the complex plane, calculated as the square root of the sum of squares of its real and imaginary parts.
   
• Complex Plane: A two-dimensional plane used to represent complex numbers, with the real part on the x-axis and the imaginary part on the y-axis.