Square Root of -15
2026-02-28 12:07 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-15

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Last updated on December 15, 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 concept of square roots is used in fields such as vehicle design, finance, etc. Here, we will discuss the square root of -15.

What is the Square Root of -15?

The square root is the inverse of the square of the number.

Since -15 is a negative number, its square root is not a real number.

The square root of -15 is expressed in terms of imaginary numbers.

In the complex number system, it is expressed as √(-15) = √(15) * i, where i is the imaginary unit with the property that i² = -1.

√(15) is approximately 3.87298, thus √(-15) = 3.87298i.

Understanding the Square Root of -15

The square root of a negative number is defined only in the complex number system.

Here, we express the square root of -15 using the imaginary unit 'i'.

The process involves recognizing that the square root of a negative number can be simplified using the property of i, where i² = -1.

Square Root of -15 in Exponential Form

In exponential form, the square root of -15 can be written using the property of complex numbers.

It is expressed as (-15)(1/2) = (15)(1/2) * i.

This indicates the square root of 15 multiplied by the imaginary unit i, giving us approximately 3.87298i.

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Calculating the Square Root of -15

Since -15 does not have a real square root, we calculate its square root in terms of the imaginary unit i. The steps are:

Step 1: Recognize that the square root of -15 can be expressed as √15 * i.

Step 2: Calculate √15, which is approximately 3.87298.

Step 3: Multiply √15 by i to get the final result: 3.87298i.

Conceptualizing Imaginary Numbers

Imaginary numbers arise when taking square roots of negative numbers.

The imaginary unit i is defined such that i² = -1.

Hence, when dealing with square roots of negative numbers, such as -15, we use i to express the result.

This expands our number system to include complex numbers of the form a + bi, where a and b are real numbers.

Common Mistakes and How to Avoid Them in Understanding the Square Root of -15

Students often make mistakes when dealing with square roots of negative numbers, particularly in recognizing the role of imaginary numbers.

Let's explore some common errors and how to avoid them.

Problem 1

What is the value of (√(-15))²?

Okay, lets begin

The value is -15.

Explanation

When we square the square root of -15, we should return to the original number.

(√(-15))² = (√(15) * i)²

= 15 * i²

= 15 * (-1)

= -15.

Well explained 👍

Problem 2

Express the square root of -15 in terms of real and imaginary parts.

Okay, lets begin

The real part is 0, and the imaginary part is 3.87298i.

Explanation

The square root of -15 is expressed as 0 + 3.87298i, where 0 is the real part and 3.87298i is the imaginary part.

Well explained 👍

Problem 3

What is the result of multiplying √(-15) by √(-15)?

Okay, lets begin

The result is -15.

Explanation

Multiplying √(-15) by itself gives (√(-15))², which equals -15, as shown in the calculation of the square root squared.

Well explained 👍

Problem 4

How do you write the square root of -15 using the imaginary unit?

Okay, lets begin

It is written as 3.87298i.

Explanation

The square root of -15 is expressed in terms of i, the imaginary unit, as √15 * i, which is approximately 3.87298i.

Well explained 👍

Problem 5

Find the magnitude of the complex number representing the square root of -15.

Okay, lets begin

The magnitude is 3.87298.

Explanation

The magnitude of a complex number a + bi is calculated as √(a² + b²).

For 0 + 3.87298i, the magnitude is √(0² + (3.87298)²) = 3.87298.

Well explained 👍

FAQ on Square Root of -15

1.Can the square root of a negative number be a real number?

No, the square root of a negative number cannot be a real number.

It is expressed as an imaginary number using the imaginary unit i.

2.What is the imaginary unit?

The imaginary unit, denoted as i, is defined such that i² = -1.

It is used to express the square roots of negative numbers.

3.How do you express the square root of -15 in exponential form?

In exponential form, the square root of -15 is expressed as (15)(1/2) * i.

4.What is the principal square root of -15?

The principal square root of -15 is 3.87298i, where 3.87298 is the square root of 15, and i is the imaginary unit.

5.Is there a difference between real and imaginary numbers?

Yes, real numbers are numbers without an imaginary part, while imaginary numbers are multiples of i, the square root of -1.

Important Glossaries for the Square Root of -15

  • Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. In case of negative numbers, it involves the imaginary unit i.
  • Imaginary number: A number that can be written as a real number multiplied by the imaginary unit i, where i² = -1.
  • Complex number: A number of the form a + bi, where a and b are real numbers and i is the imaginary unit.
  • Imaginary unit: Denoted as i, it is used to represent the square root of -1.
  • Magnitude: The magnitude of a complex number a + bi is √(a² + b²), representing its distance from the origin in the complex plane.

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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.