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Last updated on May 26th, 2025

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Square Root of -192

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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 the fields of complex numbers, electrical engineering, etc. Here, we will discuss the square root of -192.

Square Root of -192 for Indian Students
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What is the Square Root of -192?

The square root is the inverse of the square of the number. Since -192 is negative, its square root is not a real number. Instead, it is expressed in terms of the imaginary unit 'i', where i = √-1. Thus, the square root of -192 is expressed as √-192 or 4√12i in simplified form. Since it involves 'i', it is a complex number.

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Finding the Square Root of -192

To find the square root of a negative number, we use the concept of imaginary numbers. For -192, we separate it into its positive counterpart and the imaginary unit:

 

Step 1: Recognize the negative sign in front of 192. This indicates the presence of 'i' in the result, as i = √-1.

 

Step 2: Find the square root of the positive part, 192, separately, which is √192 = 4√12.

 

Step 3: Combine this with 'i' to express the full square root: √-192 = 4√12i.

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Square Root of 192 by Prime Factorization Method

To find the square root of 192, we first perform prime factorization:

 

Step 1: Finding the prime factors of 192: 192 = 2 x 2 x 2 x 2 x 2 x 3 = 2^5 x 3.

 

Step 2: Pair the prime factors: (2^5 x 3) can be grouped as 2^2 x 2^2 x 2 x 3.

 

Step 3: Simplify the pairs: (2^2 x 2^2) becomes 4, and the remaining 2 x 3 remains under the square root.

 

Thus, √192 = 4√12.

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Square Root of -192 by Long Division Method

The long division method is not typically used for imaginary numbers, but it can demonstrate finding the square root of the positive part:

 

Step 1: Group 192 into pairs from right to left: 92 and 1.

 

Step 2: Find a number whose square is less than or equal to 1. This is 1, with a remainder of 0.

 

Step 3: Bring down 92, making the new dividend 92. Double the previous quotient (1) to get the new divisor, 2.

 

Step 4: Find the largest digit 'n' such that 2n x n ≤ 92. In this case, n is 4 because 24 x 4 = 96.

 

Step 5: Continue the division to find √192.

 

Step 6: The result is 4√12, which for -192 includes 'i': 4√12i.

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Square Root of -192 by Approximation Method

The approximation method can be applied to the positive part of -192.

 

Step 1: Find two perfect squares between which 192 lies. These are 169 (13^2) and 196 (14^2).

 

Step 2: Since √192 is between √169 and √196, it lies between 13 and 14.

 

Step 3: Use the approximation formula: (192 - 169) / (196 - 169) = 23/27 ≈ 0.85.

 

Step 4: Add this to the lower square root: 13 + 0.85 = 13.85.

 

Step 5: For -192, include 'i': the approximation becomes approximately 13.85i.

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Common Mistakes and How to Avoid Them in the Square Root of -192

Students often make errors when dealing with square roots of negative numbers, especially forgetting about the imaginary unit 'i'. Let's look at some common mistakes:

Mistake 1

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Forgetting about the imaginary unit 'i'

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When dealing with negative numbers, students must remember to include 'i', the imaginary unit.

For example, √-192 should be expressed as 4√12i, not just 4√12.

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Square Root of -192 Examples

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Problem 1

Can you help Alex find the magnitude of an impedance if its value is given as √-192 ohms?

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The magnitude of the impedance is 13.85 ohms.

Explanation

Magnitude ignores the imaginary unit in terms of absolute value.

Therefore, the magnitude is √192 calculated as 13.85.

This is the positive root of the squared real part.

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Problem 2

A capacitor has a reactance of -192 ohms. What is the reactance in terms of imaginary numbers?

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The reactance is 4√12i ohms.

Explanation

Convert the negative reactance by including the imaginary unit: √-192 = 4√12i.

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Problem 3

Calculate √-192 x 2.

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Result is 8√12i.

Explanation

First, find the square root of -192, which is 4√12i, and then multiply by 2: 4√12i x 2 = 8√12i.

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Problem 4

What is the imaginary square root of (192 - 24)?

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The square root is 12√i.

Explanation

First, find the difference: 192 - 24 = 168.

Then, √-168 = √(168)i = 12√i, simplifying to 12√i.

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Problem 5

Find the complex conjugate of a number if its imaginary part is √-192.

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The complex conjugate is -4√12i.

Explanation

The complex conjugate changes the sign of the imaginary part: if the imaginary part is 4√12i, the conjugate is -4√12i.

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FAQ on Square Root of -192

1.What is √-192 in its simplest form?

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2.What are the imaginary units and their significance?

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3.Is it possible to have a real number as a square root of -192?

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4.How do complex numbers relate to square roots of negative numbers?

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5.Can √-192 be used in real-world applications?

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6.How does learning Algebra help students in India make better decisions in daily life?

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7.How can cultural or local activities in India support learning Algebra topics such as Square Root of -192?

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8.How do technology and digital tools in India support learning Algebra and Square Root of -192?

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9.Does learning Algebra support future career opportunities for students in India?

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Important Glossaries for the Square Root of -192

  • Imaginary unit: The fundamental imaginary unit 'i' is defined as √-1. It is used to handle square roots of negative numbers.

 

  • Complex number: A number that comprises a real and an imaginary part, typically expressed in the form a + bi where 'a' is the real part and 'bi' is the imaginary part.

 

  • Real part: The component of a complex number that does not include the imaginary unit 'i'. For example, in 3 + 4i, the real part is 3.

 

  • Imaginary part: The component of a complex number that includes the imaginary unit 'i'. For example, in 3 + 4i, the imaginary part is 4i.

 

  • Magnitude: The magnitude of a complex number is its absolute value, calculated as the square root of the sum of the squares of its real and imaginary parts.
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About BrightChamps in India

At BrightChamps, we see algebra as more than just symbols—it opens doors to endless opportunities! Our mission is to help children all over India develop vital math skills, focusing today on the Square Root of -192 with special attention to understanding square roots—in a way that’s engaging, lively, and easy to follow. Whether your child is calculating the speed of a passing train, keeping scores during a cricket match, or managing pocket money for the latest gadgets, mastering algebra gives them the confidence needed for everyday situations. Our interactive lessons keep learning simple and fun. As kids in India have varied learning styles, we personalize our approach to match each child. From the busy markets of Mumbai to Delhi’s vibrant streets, BrightChamps brings math to life, making it relatable and exciting throughout India. Let’s make square roots a joyful part of every child’s math journey!
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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.

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Fun Fact

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

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