Square Root of -164
2026-02-28 08:06 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-164

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Last updated on August 5, 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 and finance. Here, we will discuss the square root of -164.

What is the Square Root of -164?

The square root is the inverse of the square of a number. Since -164 is a negative number, its square root involves imaginary numbers. The square root of -164 is expressed in both radical and exponential form. In the radical form, it is expressed as √(-164), whereas (-164)^(1/2) in the exponential form. The square root of a negative number is an imaginary number, which can be expressed as i√164, where i is the imaginary unit with the property that i² = -1.

Finding the Square Root of -164

To find the square root of a negative number like -164, we need to recognize that it involves the imaginary unit i. The prime factorization and long division methods do not apply directly to negative numbers since they yield real numbers. However, we can find the square root of the positive counterpart and then multiply it by i. Let's explore the steps:

1. Find the square root of 164 using the usual methods.

2. Multiply the result by i to account for the negative sign.

Square Root of 164 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Here’s how 164 is broken down into its prime factors:

Step 1: Finding the prime factors of 164 Breaking it down, we get 2 x 2 x 41: 2² x 41

Step 2: Pair the prime factors. Since 164 is not a perfect square, the digits of the number can’t be grouped into pairs for complete pairs.

Thus, the square root of 164 is expressed as 2√41, and for -164 as i(2√41).

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Square Root of 164 by Long Division Method

The long division method is typically used for finding the square roots of positive non-perfect square numbers. Here’s how to find the square root of 164 using the long division method, then apply the result to -164:

Step 1: Group the numbers from right to left. For 164, group as 64 and 1.

Step 2: Find n whose square is 1. Here, n is 1 because 1 x 1 = 1. Subtract 1 from 1, resulting in a remainder of 0.

Step 3: Bring down 64, the new dividend. Add the previous divisor 1 to itself to get 2, the new divisor.

Step 4: Find n such that 2n x n ≤ 64. Here, n is 8 because 28 x 8 = 224, and 224 is less than 640.

Step 5: Subtract 224 from 640 to get a remainder of 416 and continue the process to get more decimal places. Once you find the square root of 164, multiply the result by i for -164.

Square Root of 164 by Approximation Method

The approximation method is another way to find square roots. Here’s how to find the square root of 164 using the approximation method:

Step 1: Identify the closest perfect squares to 164. The nearest perfect squares are 144 (12²) and 169 (13²). Thus, √164 is between 12 and 13.

Step 2: Use the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying it: (164 - 144) / (169 - 144) = 20 / 25 = 0.8 Add the decimal to the lower bound: 12 + 0.8 = 12.8 For -164, multiply the result by i to get i(12.8).

Common Mistakes and How to Avoid Them in the Square Root of -164

Students often make mistakes while finding square roots, such as forgetting about the imaginary unit when dealing with negative numbers. Let’s explore some common mistakes in detail.

Problem 1

If Max wants to find the result of i times the square root of 138, what will it be?

Okay, lets begin

The result is approximately 11.75i.

Explanation

To find the result, calculate the square root of 138, which is approximately 11.75, and multiply by i.

Well explained 👍

Problem 2

A square has an area of -164 square units. What is the side length in terms of imaginary numbers?

Okay, lets begin

The side length is i√164.

Explanation

The side length of a square is the square root of its area.

For negative areas, use i to denote the imaginary part: i√164.

Well explained 👍

Problem 3

Calculate i√164 x 3.

Okay, lets begin

The result is approximately 38.4i.

Explanation

First, find the square root of 164, which is approximately 12.8, then multiply by 3 and add the imaginary unit: 12.8 x 3 = 38.4, thus 38.4i.

Well explained 👍

Problem 4

What is the square root of (138 - 2) in terms of imaginary numbers?

Okay, lets begin

The square root is approximately 11.66i.

Explanation

First, find the square root of (138 - 2) = 136, which is approximately 11.66.

Thus, the square root of the negative is 11.66i.

Well explained 👍

Problem 5

If a rectangle has a length of i√138 units and a width of 38 units, what is its perimeter in terms of imaginary numbers?

Okay, lets begin

The perimeter is approximately 77.48 + 11.66i units.

Explanation

Perimeter of the rectangle = 2 × (length + width), where length = i√138 ≈ 11.66i, width = 38.

Thus, the perimeter = 2 × (11.66i + 38) = 77.48 + 23.32i.

Well explained 👍

FAQ on Square Root of -164

1.What is the square root of -164 in its simplest form?

The simplest form of the square root of -164 is i√164, where i is the imaginary unit.

2.What are the factors of 164?

The factors of 164 are 1, 2, 4, 41, 82, and 164.

3.Calculate the square of -164.

The square of -164 is 26896, as (-164)² = 26896.

4.Is 164 a prime number?

No, 164 is not a prime number, as it has more than two factors.

5.Is -164 divisible by any numbers?

-164 is divisible by its factors: 1, 2, 4, 41, 82, and 164.

Important Glossaries for the Square Root of -164

  • Imaginary Unit: The imaginary unit i is defined such that i² = -1. It is 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, the square root involves the imaginary unit i.
  • Prime Factorization: Breaking down a number into the product of its prime factors. For example, 164 = 2² x 41.
  • Perfect Square: A number that is the square of an integer. For example, 144 is a perfect square since 12² = 144.
  • Approximation: The process of finding a close estimate of a number, often used for non-perfect squares.

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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.