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Last updated on May 26th, 2025

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Square Root of -39

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If a number is multiplied by itself, the result is a square. The inverse of squaring a number is finding its square root. The concept of square roots is used in various fields such as engineering, physics, and mathematics. Here, we will discuss the square root of -39.

Square Root of -39 for Canadian Students
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What is the Square Root of -39?

The square root is the inverse operation of squaring a number. Since -39 is a negative number, its square root is not a real number. The square root of -39 is expressed in terms of imaginary numbers. In radical form, it is expressed as √(-39), while in exponential form it is (−39)^(1/2). The square root of -39 is an imaginary number and can be written as 3√13 * i, where i is the imaginary unit with the property that i^2 = -1.

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Finding the Square Root of -39

The square root of a negative number is not real, so we use the concept of imaginary numbers. For non-negative numbers, methods like prime factorization, long division, and approximation are used, but for negative numbers, we directly express the square root in terms of i. Here is how to express √(-39):

 

1. Express the number with the negative sign separately: √(-39) = √(39) * √(-1).

2. Simplify to get the imaginary number: √(39) * i.

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Square Root of -39 by Imaginary Number Concept

The concept of imaginary numbers is essential for finding the square root of negative numbers:

 

Step 1: Identify the negative number under the radical: √(-39).

 

Step 2: Separate the negative sign: √(39) * √(-1).

 

Step 3: Simplify using the imaginary unit: √(39) * i.

 

Step 4: Simplify further to get the square root of 39 in its simplest form: 3√13 * i.

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Understanding Imaginary Numbers

Imaginary numbers are used when dealing with the square roots of negative numbers. The imaginary unit, denoted as i, is defined by the property i^2 = -1. Thus, when calculating the square root of -39, we express it as an imaginary number:

 

1. Separate the negative component: √(-39) = √(39) * i.

2. Calculate √39 approximately: √39 ≈ 6.244.

3. Combine with i: √(-39) = 6.244 * i.

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Applications of Imaginary Numbers

Imaginary numbers, although not used in everyday real-world counting, have significant applications in advanced fields:

 

1. Electrical Engineering: Used in analyzing AC circuits and impedance.

2. Quantum Physics: Essential in wave functions and complex numbers.

3. Control Systems: Applied in stability analysis and system response.

4. Signal Processing: Utilized in Fourier transforms and filtering.

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Common Mistakes and How to Avoid Them in the Square Root of -39

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

Mistake 1

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Ignoring the Imaginary Unit

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It is crucial to remember that the square root of a negative number involves the imaginary unit i. Forgetting to include i can lead to incorrect results.

For instance, √(-39) should be expressed as 6.244 * i, not just 6.244.

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Square Root of -39 Examples

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Problem 1

Can you help Max find the imaginary part of the number if the expression is given as 4 + √(-39)?

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The imaginary part of the expression is 6.244i.

Explanation

The expression 4 + √(-39) can be rewritten using the imaginary unit: 4 + 6.244i.

The imaginary part is 6.244i.

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Problem 2

If a function f(x) = √(-x), what is f(39)?

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f(39) = 6.244i.

Explanation

Substitute x = 39 into the function: f(39) = √(-39).

The square root of -39 is 6.244i, so f(39) = 6.244i.

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Problem 3

Calculate 2 * √(-39).

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The result is 12.488i.

Explanation

First, find the square root of -39, which is 6.244i.

Then multiply by 2: 2 * 6.244i = 12.488i.

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Problem 4

What is the real part of the expression 7 - √(-39)?

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The real part is 7.

Explanation

The expression 7 - √(-39) is written as 7 - 6.244i.

The real part is 7.

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Problem 5

If z = √(-39), what is the modulus of z?

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The modulus of z is 6.244.

Explanation

The modulus of an imaginary number bi is |b|.

Since z = 6.244i, its modulus is 6.244.

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FAQ on Square Root of -39

1.What is √(-39) in terms of imaginary numbers?

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2.Why can't we find a real square root for -39?

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3.How do you represent √(-39) using the imaginary unit?

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4.What is the principal square root of a negative number?

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5.Can imaginary numbers be used in real-world applications?

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

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7.How can cultural or local activities in Canada support learning Algebra topics such as Square Root of -39?

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8.How do technology and digital tools in Canada support learning Algebra and Square Root of -39?

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

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Important Glossaries for the Square Root of -39

  • Imaginary Number: A number of the form bi, where i is the imaginary unit and b is a real number. For example, 6.244i is an imaginary number.
     
  • Imaginary Unit: Denoted by i, it is defined by the property i^2 = -1.
     
  • Complex Number: A number of the form a + bi, where a and b are real numbers and i is the imaginary unit. Principal
     
  • Square Root: The non-negative square root of a number, extended to the imaginary unit for negative numbers.
     
  • Modulus: The modulus of a complex number a + bi is √(a^2 + b^2). For an imaginary number bi, it is |b|.
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About BrightChamps in Canada

At BrightChamps, we know algebra is more than just symbols—it’s a key to open many doors! Our aim is to support kids across Canada in grasping important math skills, such as today’s focus on the Square Root of -39, highlighting square roots in a fun, engaging, and easy-to-understand way. Whether your child is measuring how fast a roller coaster moves at Canada’s Wonderland, tracking hockey scores, or planning their allowance for the latest gadgets, mastering algebra helps build their confidence for daily tasks. Our hands-on lessons make learning enjoyable and straightforward. Since kids in Canada learn in diverse ways, we tailor lessons to their individual style. From Toronto’s busy streets to British Columbia’s stunning landscapes, BrightChamps brings math to life across Canada. Let’s make square roots a fun part of your child’s math experience!
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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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