Square Root of -144
2026-02-28 13:44 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-144

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Last updated on August 5, 2025

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in fields such as electrical engineering, quantum physics, etc. Here, we will discuss the square root of -144.

What is the 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.

Understanding the Square Root of -144

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

Square Root of -144 Using 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.

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Applications of Imaginary Numbers

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.

Common Mistakes and How to Avoid Them in Finding Square Roots of Negative Numbers

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

Forgetting the Imaginary Unit 'i'

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.

Problem 1

What is the square of the square root of -144?

Okay, lets begin

The square is -144.

Explanation

Since the square root of -144 is 12i, squaring it gives (12i)² = 144i² = 144(-1) = -144.

Well explained 👍

Problem 2

If x = √(-144), what is x²?

Okay, lets begin

x² is -144.

Explanation

Given x = √(-144), we have x = 12i.

Therefore, x² = (12i)² = 144i² = 144(-1) = -144.

Well explained 👍

Problem 3

Calculate 3 times the square root of -144.

Okay, lets begin

36i

Explanation

First, find the square root of -144, which is 12i.

Then multiply by 3: 3 * 12i = 36i.

Well explained 👍

Problem 4

What is the product of √(-144) and √(-1)?

Okay, lets begin

-12

Explanation

√(-144) is 12i and √(-1) is i.

The product is 12i * i = 12i² = 12(-1) = -12.

Well explained 👍

Problem 5

Express √(-144) in polar form.

Okay, lets begin

12cis(π/2)

Explanation

The polar form of a complex number is given by r(cisθ), where r is the magnitude and θ is the argument.

For 12i, the magnitude is 12, and the argument is π/2.

Therefore, it is 12cis(π/2).

Well explained 👍

FAQ on Square Root of -144

1.What is √(-144) in its simplest form?

The simplest form of √(-144) is 12i, using the imaginary unit i, where i² = -1.

2.What is the imaginary unit 'i'?

The imaginary unit 'i' is defined as √(-1). It is used to express the square roots of negative numbers.

3.How is the square root of a negative number used in real life?

Imaginary numbers, including square roots of negative numbers, are used in electrical engineering, quantum physics, signal processing, and control systems.

4.Can √(-144) be a real number?

No, the square root of a negative number is not a real number; it is an imaginary number.

5.Why is √(-144) written as 12i?

The positive square root of 144 is 12, and the negative sign is expressed using the imaginary unit i, resulting in 12i.

Important Glossaries for the Square Root of -144

  • 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.

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.