Last updated on May 26th, 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.
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.
What is the square of the square root of -144?
The square is -144.
Since the square root of -144 is 12i, squaring it gives (12i)² = 144i² = 144(-1) = -144.
If x = √(-144), what is x²?
x² is -144.
Given x = √(-144), we have x = 12i.
Therefore, x² = (12i)² = 144i² = 144(-1) = -144.
Calculate 3 times the square root of -144.
36i
First, find the square root of -144, which is 12i.
Then multiply by 3: 3 * 12i = 36i.
What is the product of √(-144) and √(-1)?
-12
√(-144) is 12i and √(-1) is i.
The product is 12i * i = 12i² = 12(-1) = -12.
Express √(-144) in polar form.
12cis(π/2)
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).
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.
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