Square Root of -216
2026-02-28 10:58 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-216

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Last updated on August 5, 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.

What is 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.

Finding the Square Root of -216

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'

Square Root of -216 by Expressing in Terms of '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.

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Square Root of -216 by Calculating the Positive Part

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.

Square Root of -216 by Combining with 'i'

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.

Common Mistakes and How to Avoid Them in the Square Root of -216

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:

Problem 1

What is the square root of -256?

Okay, lets begin

The square root is 16i.

Explanation

For -256, we first calculate √256 = 16.

Then, multiply by 'i' to account for the negative, resulting in 16i.

Well explained 👍

Problem 2

How do you express the square root of -81?

Okay, lets begin

9i

Explanation

The square root of 81 is 9.

Combining with 'i', the square root of -81 is 9i.

Well explained 👍

Problem 3

Calculate √(-49) x 3.

Okay, lets begin

The result is 21i.

Explanation

First, find the square root of -49, which is 7i.

Then multiply by 3 to get 21i.

Well explained 👍

Problem 4

What is the square root of (-64 + 16)?

Okay, lets begin

The square root is 8i.

Explanation

First, simplify (-64 + 16) to -48.

Then, calculate √48 ≈ 6.93.

Finally, express it as 6.93i.

Well explained 👍

Problem 5

If the length of a side of a square is √(-36), what is its area?

Okay, lets begin

The area is -36 square units.

Explanation

Since the side is 6i, the area is (6i)² = -36.

Well explained 👍

FAQ on Square Root of -216

1.What is √(-216) in its simplest form?

The simplest form of √(-216) is 14.7i, where 14.7 is the approximate square root of 216.

2.What are the factors of 216?

Factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.

3.Calculate the square of 216.

The square of 216 is 46656, calculated by multiplying 216 by itself.

4.Is -216 a prime number?

No, -216 is not a prime number. Prime numbers are positive and have only two distinct positive divisors: 1 and themselves.

5.What is the cube root of -216?

The cube root of -216 is -6, since (-6)³ = -216.

Important Glossaries for the Square Root of -216

  • Complex Number: A number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit.
  • Imaginary Unit 'i': Defined as the square root of -1, used to express the square roots of negative numbers.
  • Square Root: The square root of a number x is a number y such that y² = x. For negative numbers, it involves 'i'.
  • Irrational Number: A real number that cannot be expressed as a simple fraction. Square roots of non-perfect squares are often irrational.
  • Approximation: The process of finding a value that is close enough to the correct value, typically within an acceptable range of error.

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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.