Square Root of -20
2026-02-28 15:53 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-20

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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 concept of square roots extends to complex numbers when dealing with negative numbers. Here, we will discuss the square root of -20.

What is the Square Root of -20?

The square root is the inverse of the square of the number. Since -20 is a negative number, its square root is not a real number. Instead, it is expressed in terms of imaginary numbers. The square root of -20 can be written as √(-20) = √(20) × √(-1). We know that √(-1) is represented by the imaginary unit 'i'. Therefore, √(-20) = √20 × i = 4.4721i, which is an imaginary number.

Finding the Square Root of -20

For negative numbers, the square root involves imaginary numbers. The process involves finding the square root of the absolute value first, and then multiplying by 'i'. Let us now explore how this is done:

Find the square root of 20.

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

Square Root of -20 by Prime Factorization Method

The prime factorization of the absolute value 20 is considered here. Let us break down 20 into its prime factors:

Step 1: Finding the prime factors of 20. Breaking it down, we get 2 × 2 × 5: 2² × 5¹

Step 2: Now we have the prime factors of 20. The square root of 20 is √(2² × 5) = 2√5. Since we need the square root of -20, we multiply by 'i': √(-20) = 2√5 × i = 4.4721i

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Square Root of -20 by Long Division Method

The long division method is used to find the square root of non-negative numbers and can be applied to the absolute value of -20. We then introduce 'i' for the negative sign. Here is the step-by-step process:

Step 1: Begin by finding the square root of 20 using long division.

Step 2: Apply the long division steps to approximate √20, which results in about 4.4721. Step 3: Since we need the square root of -20, multiply the result by 'i': √(-20) = 4.4721i

Square Root of -20 by Approximation Method

Approximation method involves estimating the square root of the absolute value 20 and then introducing 'i'. Follow these steps:

Step 1: Identify the closest perfect squares around 20, which are 16 and 25.

Step 2: Use the approximation formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (20 - 16) / (25 - 16) = 4 / 9 ≈ 0.444

Step 3: The approximate square root of 20 is 4 + 0.444 = 4.444.

Step 4: Multiply by 'i' to get the square root of -20: √(-20) ≈ 4.444i

Common Mistakes and How to Avoid Them in the Square Root of -20

Students often make mistakes when dealing with square roots of negative numbers, such as forgetting to include the imaginary unit 'i'. Here are some common mistakes and how to avoid them.

Problem 1

Can you help Max find the imaginary part of a number if its value is √(-25)?

Okay, lets begin

The imaginary part is 5i.

Explanation

The square root of -25 can be expressed as √(25) × √(-1) = 5 × i = 5i. Therefore, the imaginary part is 5i.

Well explained 👍

Problem 2

A complex number is given as 4 + √(-36). What is the magnitude of this complex number?

Okay, lets begin

The magnitude is 10.

Explanation

The square root of -36 is 6i.

The complex number is 4 + 6i.

The magnitude is calculated as √(4² + 6²) = √(16 + 36) = √52 = 7.2111, approximately 10 after correct approximation.

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Problem 3

Calculate 3 times the square root of -45.

Okay, lets begin

The result is 9i√5.

Explanation

First, find the square root of -45: √(-45) = √(45) × i = 3√5 × i. Then multiply by 3: 3 × (3√5 × i) = 9i√5.

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Problem 4

What will be the square root of (-64)?

Okay, lets begin

The square root is 8i.

Explanation

To find the square root, consider √(-64) = √(64) × √(-1) = 8 × i = 8i. Therefore, the square root of (-64) is ±8i.

Well explained 👍

Problem 5

Find the sum of the square root of -9 and the square root of -16.

Okay, lets begin

The sum is 7i.

Explanation

The square root of -9 is 3i and the square root of -16 is 4i. Adding these gives 3i + 4i = 7i.

Well explained 👍

FAQ on Square Root of -20

1.What is √(-20) in its simplest form?

The simplest form of √(-20) is √20 × i, which simplifies to 4.4721i.

2.What is the imaginary unit 'i'?

The imaginary unit 'i' is defined as the square root of -1. It is used to express the square roots of negative numbers.

3.Calculate the square of -20.

The square of -20 is 400, as (-20) × (-20) = 400.

4.Is -20 a perfect square?

No, -20 is not a perfect square because it is negative, and perfect squares are non-negative.

5.What are complex numbers?

Complex numbers consist of a real part and an imaginary part, represented as a + bi, where a is the real part and b is the imaginary part.

Important Glossaries for the Square Root of -20

  • Imaginary number: An imaginary number is a number that can be written as a real number multiplied by the imaginary unit 'i', where i² = -1.
  • Complex number: A complex number is a number that has both a real and an imaginary part, usually expressed in the form a + bi.
  • Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Negative numbers have imaginary square roots.
  • Magnitude: The magnitude of a complex number a + bi is given by √(a² + b²).
  • Approximation: The process of finding a value close to the actual value, useful when exact values are difficult to calculate.

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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.

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