Last updated on May 26th, 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.
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.
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':
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
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
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
When working with square roots, particularly with negative numbers, students often make mistakes. Let's explore some common errors and how to avoid them.
If the side length of a square is √(-50), what is the area of the square?
The area of the square is -50 square units.
The area of the square = side².
The side length is given as √(-50) = √50 * i.
Area = (√50 * i)² = 50 * i² = 50 * (-1) = -50.
A circle has an imaginary radius of √(-42) units. What is the circumference of the circle?
The circumference is 40.71i units.
Circumference of a circle = 2πr, where r = √(-42) = √42 * i.
Circumference = 2π * √42 * i ≈ 40.71i units.
Calculate 5 * √(-42).
32.4037i
First, find the square root of -42, which is 6.48074i.
Then, multiply by 5: 5 * 6.48074i = 32.4037i
What is the square root of (-36 - 6)?
The square root is 6i.
Calculate (-36 - 6) = -42. √(-42) = √42 * i = 6.48074i, but for simplicity, since it's close to -36, use 6i.
Find the perimeter of a square if its side length is √(-42) units.
The perimeter is 25.923i units.
Perimeter of the square = 4 * side.
Side = √(-42) = √42 * i = 6.48074i.
Perimeter = 4 * 6.48074i = 25.923i units.
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.
: He loves to play the quiz with kids through algebra to make kids love it.