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Last updated on April 9th, 2025

Square Root of -108

Q: 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.

Q: What is the principal square root of -108?

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

Q: 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.

Q: Is -108 a perfect square?

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

Q: 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.

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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 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. $square root of minus 108$

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.

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

Square Root of -108 Examples

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Problem 1

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

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Explanation

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)?

Explanation

Problem 3

Calculate √(-108) x 5.

Explanation

Problem 4

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

Explanation

Problem 5

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

Explanation

FAQ on Square Root of -108

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

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

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

4.Is -108 a perfect square?

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

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

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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.