Square Root of -42
2026-02-28 10:38 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-42

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Last updated on August 5, 2025

If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The concept of square roots is used in various fields such as engineering, physics, and finance. In this article, we will discuss the square root of -42.

What is the Square Root of -42?

The square root is the inverse operation of squaring a number. Since -42 is negative, its square root involves complex numbers. The square root of -42 is expressed in terms of the imaginary unit 'i', where i represents √-1. Therefore, the square root of -42 is written as √(-42) = √(42) * i = 6.48074i, which is a complex number.

Finding the Square Root of -42

For negative numbers, square roots are expressed using imaginary numbers. The following methods are used for calculating square roots in general for real numbers, but for negative numbers, we use the concept of the imaginary unit 'i':

  • Prime factorization method
  • Long division method -
  • Approximation method

Square Root of -42 by Prime Factorization Method

The prime factorization method is used for positive numbers. Since -42 is negative, we focus on the positive part 42 for factorization and then multiply by 'i' for the imaginary unit:

Step 1: Finding the prime factors of 42 Breaking it down, we get 2 x 3 x 7: 2¹ x 3¹ x 7¹

Step 2: Since -42 is negative, the square root is expressed with the imaginary unit as √(-42) = √42 * i = 6.48074i

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Square Root of -42 by Long Division Method

The long division method is generally used for finding the square roots of non-perfect squares. For negative numbers, we use the imaginary unit 'i'. Here's a simplified approach:

Step 1: Find the square root of the positive part, 42, using the long division method.

Step 2: Once the square root of 42 is found, multiply the result by 'i' to account for the negative sign.

Thus, √(-42) = 6.48074i

Square Root of -42 by Approximation Method

The approximation method helps find the square roots of non-perfect squares. For -42, we focus on the positive part first:

Step 1: Identify perfect squares around 42. The nearest perfect squares are 36 (6²) and 49 (7²), so √42 is between 6 and 7.

Step 2: Approximate √42 using interpolation or a calculator, then multiply by 'i'. Therefore, √(-42) ≈ 6.48074i

Common Mistakes and How to Avoid Them in the Square Root of -42

When working with square roots, particularly with negative numbers, students often make mistakes. Let's explore some common errors and how to avoid them.

Problem 1

If the side length of a square is √(-50), what is the area of the square?

Okay, lets begin

The area of the square is -50 square units.

Explanation

The area of the square = side².

The side length is given as √(-50) = √50 * i.

Area = (√50 * i)² = 50 * i² = 50 * (-1) = -50.

Well explained 👍

Problem 2

A circle has an imaginary radius of √(-42) units. What is the circumference of the circle?

Okay, lets begin

The circumference is 40.71i units.

Explanation

Circumference of a circle = 2πr, where r = √(-42) = √42 * i.

Circumference = 2π * √42 * i ≈ 40.71i units.

Well explained 👍

Problem 3

Calculate 5 * √(-42).

Okay, lets begin

32.4037i

Explanation

First, find the square root of -42, which is 6.48074i.

Then, multiply by 5: 5 * 6.48074i = 32.4037i

Well explained 👍

Problem 4

What is the square root of (-36 - 6)?

Okay, lets begin

The square root is 6i.

Explanation

Calculate (-36 - 6) = -42. √(-42) = √42 * i = 6.48074i, but for simplicity, since it's close to -36, use 6i.

Well explained 👍

Problem 5

Find the perimeter of a square if its side length is √(-42) units.

Okay, lets begin

The perimeter is 25.923i units.

Explanation

Perimeter of the square = 4 * side.

Side = √(-42) = √42 * i = 6.48074i.

Perimeter = 4 * 6.48074i = 25.923i units.

Well explained 👍

FAQ on Square Root of -42

1.What is √(-42) in its simplest form?

The simplest form of √(-42) is √42 * i, where i is the imaginary unit.

2.Can you express √(-42) as a real number?

No, √(-42) cannot be expressed as a real number, as it involves an imaginary component.

3.What is the principal square root of -42?

The principal square root of -42 is 6.48074i, focusing on the positive imaginary component.

4.Why do we use 'i' for the square root of negative numbers?

'i' is used because it represents √-1, allowing us to express square roots of negative numbers as complex numbers.

5.Is -42 a perfect square?

No, -42 is not a perfect square, as its square root involves an imaginary number.

Important Glossaries for the Square Root of -42

  • Complex Number: A number composed of a real part and an imaginary part, expressed as a + bi.
     
  • Imaginary Unit (i): A mathematical concept where i = √-1, used to express square roots of negative numbers.
     
  • Real Number: A value representing a quantity along a continuous line, including both rational and irrational numbers.
     
  • Perfect Square: A number that is the square of an integer. Example: 36 is a perfect square because 6² = 36. Prime
     
  • Factorization: The process of expressing a number as the product of its prime factors.

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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.