Square Root of -108
2026-02-28 01:34 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-108

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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 concept of square roots extends into the realm of complex numbers when dealing with negative numbers. Here, we will discuss the square root of -108.

What is the Square Root of -108?

The square root is the inverse of the square of a number. When dealing with negative numbers, we use complex numbers to express the square root. The square root of -108 is expressed in terms of the imaginary unit 'i', where i = √(-1). Thus, the square root of -108 is written as √(-108) = √(108) * i. The value of √108 is approximately 10.3923, so √(-108) ≈ 10.3923i.

Finding the Square Root of -108

There are different methods to find square roots, but for negative numbers, the concept of imaginary numbers is used. Here's how to find the square root of -108:

1. Calculate the square root of the positive part, 108.

2. Multiply the result by i (the imaginary unit).

Square Root of -108 Using Imaginary Numbers

To find the square root of -108 using imaginary numbers, we start with the positive part:

Step 1: Calculate the square root of 108. The prime factorization of 108 is 2 x 2 x 3 x 3 x 3, or 2² x 3³.

Step 2: Simplify √108 = √(2² x 3² x 3) = 2 x 3 x √3 = 6√3 ≈ 10.3923.

Step 3: Multiply by the imaginary unit i: √(-108) = √108 * i = 10.3923i.

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Square Root of -108 by Long Division Method (Imaginary Part)

The long division method is typically used for approximating square roots of positive numbers. For -108, we focus on the positive part:

Step 1: Use long division to approximate √108, which we already found to be about 10.3923.

Step 2: Multiply this result by i to obtain the square root of the negative number: √(-108) = 10.3923i.

Square Root of -108 by Approximation Method

Approximation helps find the square root of the positive part, 108:

Step 1: Identify the perfect squares surrounding 108. 100 (10²) and 121 (11²) are the closest perfect squares.

Step 2: Approximate between these values: √108 is between 10 and 11. Using the approximation method, we find √108 ≈ 10.3923.

Step 3: Multiply by i for the imaginary part: √(-108) = 10.3923i.

Common Mistakes and How to Avoid Them in the Square Root of -108

Students often make mistakes when dealing with the square roots of negative numbers, especially with the involvement of imaginary numbers. Let's discuss some common mistakes and how to avoid them.

Problem 1

Can you help Max find the area of a square box if its side length is given as √(-108)?

Okay, lets begin

The area of the square is -108 square units.

Explanation

The area of the square = side².

The side length is given as √(-108).

Area of the square = (√(-108))² = -108.

Therefore, the area of the square box is -108 square units, considering complex units.

Well explained 👍

Problem 2

A complex square-shaped plot has an area of -108 square meters. What is the length of each side if it's given by √(-108)?

Okay, lets begin

The side length is approximately 10.3923i meters.

Explanation

The side length of the square is given by the square root of the area. √(-108) = 10.3923i meters, which is the length of each side, considering the imaginary component.

Well explained 👍

Problem 3

Calculate √(-108) x 5.

Okay, lets begin

51.9615i

Explanation

First, find the square root of -108, which is approximately 10.3923i. Multiply by 5: 10.3923i x 5 = 51.9615i.

Well explained 👍

Problem 4

What will be the square root of (-100 + 8)?

Okay, lets begin

The square root is approximately 10.3923i.

Explanation

Calculate the sum: (-100 + 8) = -92. Then find the square root: √(-92) = √92 * i ≈ 9.5917i.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length 'l' is √(-108) units and the width 'w' is 38 units.

Okay, lets begin

The perimeter is a complex number: 76 + 20.7846i units.

Explanation

Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√(-108) + 38) = 2 × (10.3923i + 38). = 76 + 20.7846i units.

Well explained 👍

FAQ on Square Root of -108

1.What is √(-108) in its simplest form?

The prime factorization of 108 is 2² x 3³. Thus, √108 = 6√3. Therefore, in terms of complex numbers, √(-108) = 6√3i.

2.What is the principal square root of -108?

The principal square root of -108 is the positive imaginary root, 10.3923i.

3.Can the square root of -108 be simplified further?

The square root of -108 in its simplest form is 6√3i, as it involves a complex number.

4.Is -108 a perfect square?

No, -108 is not a perfect square because it is negative, and perfect squares are non-negative integers.

5.What does the imaginary unit 'i' represent?

The imaginary unit 'i' represents the square root of -1, which is used to express square roots of negative numbers.

Important Glossaries for the Square Root of -108

  • Imaginary Number: A number that can be written as a real number multiplied by the imaginary unit 'i', which is defined by i² = -1.
     
  • Complex Number: A number that has both a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers.
     
  • Square Root: The value that, when multiplied by itself, gives the original number. For negative numbers, it involves the imaginary unit.
     
  • Perfect Square: A number that is the square of an integer. Negative numbers cannot be perfect squares.
     
  • Approximation: The process of finding a number close to the exact value, often used for irrational numbers or complex calculations.

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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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