113 LearnersLast updated on May 26th, 2025
Square Root of -42
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
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.
Mistake 1
Ignoring the Imaginary Unit
A common mistake is ignoring the imaginary unit 'i' when dealing with negative square roots. Always remember that the square root of a negative number is a complex number, involving 'i'.
For example: √(-50) = √50 * i = 7.071i.
Mistake 2
Confusing with Real Numbers
Some students attempt to find real square roots for negative numbers, which is incorrect. Negative numbers have complex square roots.
Example: √(-36) should not be simplified to a real number. The correct expression is 6i.
Mistake 3
Misplacing the Imaginary Unit
Misplacing 'i' in the expression or forgetting it altogether can lead to errors. Ensure 'i' is positioned correctly: √(-42) = √42 * i.
Mistake 4
Using Incorrect Factorization
Incorrectly factorizing numbers or missing factors can result in the wrong square root.
For example, ensure correct factorization: 42 = 2 x 3 x 7.
Mistake 5
Mixing Up Square and Cube Roots
Confusing square roots with cube roots is a common mistake. Note the difference: √(-42) involves 'i', while ∛(-42) does not.
Square Root of -42 Examples
Problem 1
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.
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.
Problem 2
A circle has an imaginary radius of √(-42) units. What is the circumference of the circle?
The circumference is 40.71i units.
Explanation
Circumference of a circle = 2πr, where r = √(-42) = √42 * i.
Circumference = 2π * √42 * i ≈ 40.71i units.
Problem 3
Calculate 5 * √(-42).
32.4037i
Explanation
First, find the square root of -42, which is 6.48074i.
Then, multiply by 5: 5 * 6.48074i = 32.4037i
Problem 4
What is the square root of (-36 - 6)?
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.
Problem 5
Find the perimeter of a square if its side length is √(-42) units.
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.
FAQ on Square Root of -42
1.What is √(-42) in its simplest form?
2.Can you express √(-42) as a real number?
3.What is the principal square root of -42?
4.Why do we use 'i' for the square root of negative numbers?
5.Is -42 a perfect square?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How can cultural or local activities in United States support learning Algebra topics such as Square Root of -42?
8.How do technology and digital tools in United States support learning Algebra and Square Root of -42?
9.Does learning Algebra support future career opportunities for students in United States?
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.
Explore More algebra
About BrightChamps in United States
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.
