Square Root of -144

The square root is the inverse of the square of the number. For negative numbers, the square root involves imaginary numbers because no real number squared gives a negative result. The square root of -144 is expressed in terms of imaginary numbers, using the imaginary unit 'i', where i² = -1. Therefore, the square root of -144 can be written as √(-144) = 12i in the simplest form.

The square root of a negative number involves the use of imaginary numbers. Let us explore how we can conceptualize the square root of -144: Concept of imaginary numbers Expression in terms of 'i' Calculating with imaginary numbers

To find the square root of -144 using imaginary numbers, we separate the negative sign and the positive square root:

Step 1: Recognize that √(-144) can be expressed as √(144) * √(-1).

Step 2: Calculate √144, which is 12 since 12² = 144.

Step 3: Represent √(-1) as 'i', the imaginary unit.

Step 4: Combine the results to get 12i. Therefore, the square root of -144 is 12i.

Imaginary numbers, including the square root of a negative number, have various applications in advanced fields: Electrical engineering: Used in AC circuit analysis. Quantum physics: Helps in solving equations involving wave functions. Signal processing: Used in complex Fourier transforms. Control systems: Utilized in complex algebra for stability analysis.

Students often make mistakes when dealing with square roots of negative numbers. Here are some common errors and how to avoid them:

It's crucial to remember that the square root of a negative number involves the imaginary unit 'i'. Failing to include 'i' results in incorrect answers.

For example, √(-25) should be expressed as 5i, not 5. plain_heading7 Misunderstanding the Concept of Imaginary Numbers plain_body7 Students often confuse imaginary numbers with negative numbers or fail to grasp their applications. Teaching the concept of 'i' and its properties can help clarify these misunderstandings.

• Imaginary number: A number that can be written as a real number multiplied by the imaginary unit 'i', where i² = -1. Example: 3i.

• Complex number: A number that has both a real part and an imaginary part, usually written in the form a + bi.

• Polar form: A way of expressing complex numbers using a magnitude and an angle, often written as r(cisθ).

• Magnitude: The absolute value or modulus of a complex number, representing its distance from the origin in the complex plane.

• Argument: The angle formed by the complex number in the complex plane, measured from the positive real axis.