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, finance, and physics. Here, we will discuss the square root of -200.
The square root is the inverse of the square of the number. The number -200 is negative, and the square root of a negative number is not a real number. The square root of -200 is expressed in terms of imaginary numbers. In the radical form, it is expressed as √(-200), whereas in exponential form, it is written as (-200)^(1/2). The square root of -200 is an imaginary number and can be expressed as 10√2i, where "i" is the imaginary unit.
The prime factorization method is used for perfect square numbers. However, for negative numbers like -200, we use the concept of imaginary numbers. Let's explore the methods:
The imaginary unit "i" is defined as √(-1). Using this concept, we can express the square root of negative numbers. For -200, we simplify as follows:
Step 1: Express -200 as a product of 200 and -1: -200 = 200 × (-1)
Step 2: Take the square root of each factor: √(-200) = √(200) × √(-1)
Step 3: Simplify using the imaginary unit: √(200) = √(2 × 100) = 10√2, so √(-200) = 10√2i.
The approximation method helps find the square root of the positive equivalent of the number. Let's consider 200:
Step 1: Identify the nearest perfect squares around 200. The closest perfect squares are 196 (14^2) and 225 (15^2).
Step 2: 200 lies between 196 and 225. So, 14 < √200 < 15.
Step 3: Approximate using the average method: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square).
Using the approximation: (200 - 196) / (225 - 196) ≈ 0.14 Therefore, √200 ≈ 14.14. So, the square root of -200 is approximately 14.14i.
Imaginary numbers are used when dealing with the square roots of negative numbers. Here’s how:
1. Imaginary unit "i" represents √(-1).
2. Any negative square root can be expressed as a product of a real square root and "i".
3. For example, the square root of -200 is expressed as 10√2i.
When dealing with negative square roots, students often make errors such as ignoring the imaginary unit or incorrectly simplifying the negative square root. Let's look at some common mistakes in detail.
Can you help Max find the square root of -72?
The square root of -72 is 6√2i.
First, express -72 as a product of 72 and -1: -72 = 72 × (-1).
Then, √(-72) = √(72) × √(-1) = √(36 × 2) × i = 6√2i.
A complex number is given as 5 + √(-64). Simplify it.
5 + 8i
First, find the square root of -64: √(-64) = √(64) × √(-1) = 8i.
So, the complex number simplifies to 5 + 8i.
Calculate the product of √(-50) and 2.
The product is 10i√2.
First, find the square root of -50: √(-50) = √(50) × √(-1) = 5√2i.
Then, multiply by 2: 5√2i × 2 = 10i√2.
What is the result of (√(-81))^2?
The result is -81.
First, find the square root of -81: √(-81) = 9i.
Then, square it: (9i)^2 = 81 × (-1) = -81.
Thus, (√(-81))^2 = -81.
Find the modulus of the complex number 7 + √(-49).
The modulus is 10.
First, simplify √(-49) as 7i.
The complex number is 7 + 7i.
The modulus is √(7^2 + 7^2) = √(49 + 49) = √98 = 10.
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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