Last updated on May 26th, 2025
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.
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.
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'.
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.
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.
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.
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.
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.
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.
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.
For a negative area, the side length involves imaginary numbers.
The side length = √(-21) = √21 * i.
Calculate √(-21) * 3.
Approximately 13.749i.
First, find √21, which is approximately 4.583.
Then, multiply by 3 and the imaginary unit 'i': 4.583 * 3 * i = 13.749i.
What is the square root of (-16 - 5)?
The square root is ±i√21.
To find the square root, calculate (-16 - 5) = -21
The square root of -21 is ±√21 * i, which is ±i√21.
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.
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.
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.
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