Last updated on May 26th, 2025
When a number is multiplied by itself, the result is a square. The inverse operation is finding the square root. The concept of square roots extends to complex numbers when dealing with negative numbers. Here, we will delve into the square root of -216.
The square root of a negative number involves complex numbers, as no real number squared equals a negative number. The square root of -216 is expressed using the imaginary unit 'i', where i² = -1. Thus, the square root of -216 is written as √(-216) = √216 * i, which simplifies to approximately 14.7i. This is because √216 is approximately 14.7, and multiplying by i gives the imaginary result.
To find the square root of a negative number, we involve the imaginary unit 'i'. Let's explore the methods:
1. Expressing in terms of 'i'
2. Calculating the square root of the positive part
3. Combining with 'i'
The imaginary unit 'i' is used to represent the square root of negative numbers. For -216, the calculation is:
Step 1: Recognize that -216 can be expressed as 216 multiplied by -1.
Step 2: The square root of -216 is written as √(-216) = √(216) * √(-1).
Step 3: Since √(-1) = i, we have √(-216) = √(216) * i.
Step 4: Calculate √216. The approximate value is 14.7.
Step 5: Combine this with 'i' to get the result: 14.7i.
Finding the square root of the positive part of -216 involves:
Step 1: Calculate the square root of 216. Using approximation or estimation, √216 ≈ 14.7.
Step 2: Use the imaginary unit to represent the negative aspect. So, √(-216) = 14.7i.
Here’s how to combine the result with 'i':
Step 1: Recognize that √(-216) = √216 * i.
Step 2: Compute √216, which is approximately 14.7.
Step 3: Multiply 14.7 by 'i', giving the final square root of -216 as 14.7i.
Mistakes often occur when students deal with negative square roots. It's important to remember the role of the imaginary unit 'i'. Here are some common errors to avoid:
What is the square root of -256?
The square root is 16i.
For -256, we first calculate √256 = 16.
Then, multiply by 'i' to account for the negative, resulting in 16i.
How do you express the square root of -81?
9i
The square root of 81 is 9.
Combining with 'i', the square root of -81 is 9i.
Calculate √(-49) x 3.
The result is 21i.
First, find the square root of -49, which is 7i.
Then multiply by 3 to get 21i.
What is the square root of (-64 + 16)?
The square root is 8i.
First, simplify (-64 + 16) to -48.
Then, calculate √48 ≈ 6.93.
Finally, express it as 6.93i.
If the length of a side of a square is √(-36), what is its area?
The area is -36 square units.
Since the side is 6i, the area is (6i)² = -36.
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.