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 vehicle design, finance, etc. Here, we will discuss the square root of -196.
The square root is the inverse of the square of a number. -196 is not a perfect square, and its square root involves complex numbers because it is negative. The square root of -196 is expressed in both radical and exponential form. In the radical form, it is expressed as √(-196), whereas (-196)^(1/2) in the exponential form. The principal square root of -196 is 14i, which is an imaginary number because it involves the imaginary unit i, where i^2 = -1.
Negative numbers do not have real square roots because no real number squared gives a negative result. Instead, we use imaginary numbers to represent such square roots. The imaginary unit 'i' is used where i^2 = -1. Thus, the square root of -196 is 14i. Let's explore how to calculate and understand this:
1. Identify the positive counterpart: √(-196) = √(196) * √(-1)
2. Calculate the square root of the positive part: √196 = 14
3. Incorporate the imaginary unit: √(-196) = 14i
To calculate the square root of -196, follow these steps:
Step 1: Recognize that -196 is negative, requiring the use of the imaginary unit i.
Step 2: Find the square root of the absolute value of -196, which is 196. √196 = 14
Step 3: Combine this with the imaginary unit. Thus, √(-196) = 14i
When dealing with square roots of negative numbers, students often make errors. Some of these mistakes include: - Forgetting to use the imaginary unit 'i': Remember, the square root of a negative number involves 'i'. - Misinterpreting i^2: It's crucial to remember that i^2 = -1. - Confusing real and imaginary results: Ensure clarity between real numbers and imaginary numbers.
Imaginary numbers have applications in various fields. Here are a few examples: - Electrical engineering uses complex numbers to analyze AC circuits. - Quantum mechanics relies on complex numbers to describe wave functions. - Control theory and signal processing use imaginary numbers for system analysis.
Students may make mistakes while working with square roots of negative numbers. Here, we address some common ones:
Can you help Max find the principal square root of -49?
The principal square root of -49 is 7i.
The square root of -49 involves the imaginary unit.
First, find the square root of 49, which is 7.
Then, include the imaginary unit: √(-49) = 7i.
What is the result of multiplying √(-16) by √(-4)?
The result is 8i.
First, find the square roots: √(-16) = 4i and √(-4) = 2i.
Multiply them: (4i) * (2i) = 8i^2.
Since i^2 = -1, the result is 8(-1) = -8.
Calculate 3 times the square root of -81.
The result is 27i.
First, find the square root of -81, which is 9i.
Then multiply by 3: 3 * 9i = 27i.
If you add √(-64) and √(-36), what is the result?
The result is 10i.
First, find the square roots: √(-64) = 8i and √(-36) = 6i.
Add them: 8i + 6i = 14i.
Determine the perimeter of a rectangle if its length is √(-144) units and the width is 10 units.
The perimeter is 20 + 24i units.
Find the square root of -144, which is 12i.
The perimeter is 2 × (length + width) = 2 × (12i + 10) = 24i + 20.
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.