121 LearnersLast updated on May 26th, 2025
Square Root of -21
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in various fields such as mathematics, engineering, etc. Here, we will discuss the square root of -21.
What is the Square Root of -21?
The square root is the inverse of the square of a number. Since -21 is a negative number, its square root involves the imaginary unit 'i'. The square root of -21 is expressed in radical form as √(-21) and in exponential form as (-21)^(1/2). In terms of imaginary numbers, √(-21) = √21 * i, which is not a real number.
Finding the Square Root of -21
The square root of a negative number is not real. Instead, it involves imaginary numbers. While the prime factorization, long-division, and approximation methods are used for positive numbers, finding the square root of a negative number directly involves complex numbers and is expressed using the imaginary unit 'i'.
Square Root of -21 by Prime Factorization Method
The prime factorization method is not applicable for negative numbers when finding their square root. However, for educational purposes, if we consider the prime factorization of 21, it is 3 x 7. Thus, the square root of -21 can be expressed as i√(3 x 7) or i√21.
Square Root of -21 by Long Division Method
The long division method is not directly applicable to negative numbers. Instead, the square root of negative numbers is expressed using imaginary numbers. For -21, the square root is i√21. The long division method can be used to approximate the square root of 21 only, not for -21 directly.
Square Root of -21 by Approximation Method
The approximation method is used to find the square root of positive numbers, not negative ones. For -21, we can only approximate the square root of 21. If we approximate √21, we find it is approximately 4.583. Therefore, √(-21) is approximately 4.583i.
Common Mistakes and How to Avoid Them in the Square Root of -21
Students often make mistakes when dealing with square roots of negative numbers, such as ignoring the imaginary unit or incorrectly applying methods. Let us look at a few of these mistakes in detail.
Mistake 1
Forgetting about the imaginary unit 'i'
It is important to remember that the square root of a negative number involves the imaginary unit 'i'.
For example, the square root of -21 is √21 * i, not just √21.
Mistake 2
Not using the correct notation
Students may forget to include the imaginary unit in their expressions. When dealing with the square root of a negative number, always use 'i'.
For instance, √(-21) should be written as √21 * i.
Mistake 3
Incorrectly applying real number methods
The methods for real numbers do not apply directly to negative numbers when finding square roots. Always involve the imaginary unit when dealing with negative numbers.
Mistake 4
Confusing with absolute values
Ensure that students do not confuse the square root of a negative number with its absolute value. The square root of -21 is not √21 but √21 * i.
Mistake 5
Misunderstanding imaginary numbers
Ensure students understand that imaginary numbers are a different concept from real numbers. The square root of -21 involves 'i', representing the square root of -1.
Square Root of -21 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √(-21)?
The area of the square is -21 square units.
Explanation
The area of the square = side^2.
The side length is given as √(-21), which involves the imaginary unit.
Area of the square = (√(-21))^2 = -21 because (√(-21))^2 = -21.
Therefore, the area is -21 square units.
Problem 2
If a square-shaped plot measures -21 square feet, what would be the side length in imaginary units?
The side length is √21 * i feet.
Explanation
For a negative area, the side length involves imaginary numbers.
The side length = √(-21) = √21 * i.
Problem 3
Calculate √(-21) * 3.
Approximately 13.749i.
Explanation
First, find √21, which is approximately 4.583.
Then, multiply by 3 and the imaginary unit 'i': 4.583 * 3 * i = 13.749i.
Problem 4
What is the square root of (-16 - 5)?
The square root is ±i√21.
Explanation
To find the square root, calculate (-16 - 5) = -21
The square root of -21 is ±√21 * i, which is ±i√21.
Problem 5
Find the perimeter of a rectangle if its length 'l' is √(-21) units and the width 'w' is 4 units.
The perimeter is (8 + 2√21 * i) units.
Explanation
Perimeter of the rectangle = 2 × (length + width) = 2 × (√(-21) + 4).
Since √(-21) = √21 * i, the perimeter = 2 × (4 + √21 * i) = 8 + 2√21 * i units.
FAQ on Square Root of -21
1.What is √(-21) in its simplest form?
2.What is the imaginary unit 'i'?
3.Why can't we find the square root of a negative number using regular methods?
4.Is the square root of -21 a real number?
5.Can the square root of negative numbers be used in calculations?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of -21?
8.How do technology and digital tools in India support learning Algebra and Square Root of -21?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of -21
- Imaginary unit: The imaginary unit 'i' is defined as √(-1) and is used to express square roots of negative numbers.
- Complex number: A complex number consists of a real part and an imaginary part, often expressed as a + bi, where 'a' and 'b' are real numbers.
- Radical form: This is the expression of a square root using the radical symbol, such as √(21).
- Exponential form: This form uses exponents to express roots, such as (-21)^(1/2).
- Approximation: The process of finding a value close to the actual value, often used for non-perfect squares to find their roots approximately.
Explore More algebra
About BrightChamps in India
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.
