Square Root of -45
2026-02-28 09:52 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-45

216 Learners

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 the field of vehicle design, finance, etc. Here, we will discuss the square root of -45.

What is the Square Root of -45?

The square root is the inverse of the square of the number. Since -45 is a negative number, its square root is not a real number. Instead, it is expressed as an imaginary number. In the radical form, it is expressed as √(-45), and in terms of imaginary numbers, it is represented as 3i√5. Since it involves an imaginary unit 'i', it cannot be expressed as a rational number.

Finding the Square Root of -45

Finding the square root of a negative number involves understanding imaginary numbers. The imaginary unit 'i' is defined as √(-1). Therefore, to find the square root of -45, we express it in terms of 'i'. Let's explore this method: Imaginary unit method

Square Root of -45 by Imaginary Unit Method

To express the square root of a negative number, we use the imaginary unit 'i', where i² = -1. Let's see how -45 can be expressed:

Step 1: Recognize that -45 can be written as -1 × 45.

Step 2: The square root of -45 is √(-1 × 45) = √(-1) × √45.

Step 3: Use the imaginary unit: √(-1) = i, so we have i√45.

Step 4: Simplify √45. The prime factorization of 45 is 3 × 3 × 5 = 3² × 5. Therefore, √45 = 3√5.

Step 5: Combine the results: √(-45) = i × 3√5 = 3i√5.

Explore Our Programs

Understanding the Concept of Imaginary Numbers

Imaginary numbers are used to express the square root of negative numbers. The key component of imaginary numbers is the unit 'i', which is defined as the square root of -1. This concept allows us to find and express square roots of negative numbers that otherwise don't have real roots.

Applications of Imaginary Numbers

Imaginary numbers have practical applications in several fields, such as: - Electrical engineering: Used in alternating current (AC) circuit analysis. - Control theory: Helps in the design of control systems. - Quantum physics: Used to describe quantum states and phenomena. - Signal processing: Utilized in algorithms for processing and analyzing signals.

Common Mistakes and How to Avoid Them in the Square Root of -45

Students often make mistakes when dealing with the square root of negative numbers, particularly when involving imaginary numbers. Let's look at some common mistakes and how to avoid them.

Problem 1

Can you help Alex find the imaginary square root of -81?

Okay, lets begin

The imaginary square root is 9i.

Explanation

The square root of -81 involves the imaginary unit 'i'.

First, express -81 as -1 × 81. √(-81) = √(-1 × 81) = √(-1) × √81 = i × 9 = 9i.

Well explained 👍

Problem 2

A circuit has a negative impedance of -64 ohms. What is the imaginary square root value?

Okay, lets begin

8i ohms

Explanation

To find the square root of the negative impedance: √(-64) = √(-1 × 64) = √(-1) × √64 = i × 8 = 8i.

Well explained 👍

Problem 3

Calculate 5 times the square root of -36.

Okay, lets begin

30i

Explanation

First, find the square root of -36: √(-36) = √(-1 × 36) = i × 6 = 6i.

Next, multiply by 5: 5 × 6i = 30i.

Well explained 👍

Problem 4

What will be the imaginary square root of (-16 + 4)?

Okay, lets begin

The imaginary square root is 4i.

Explanation

Calculate the sum: -16 + 4 = -12.

Find the square root: √(-12) = √(-1 × 12) = i × √12 = i × 2√3 = 2i√3.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √(-25) units and the width ‘w’ is 5 units.

Okay, lets begin

The perimeter of the rectangle is not a real number.

Explanation

The length involves an imaginary number: √(-25) = 5i.

Perimeter = 2 × (length + width) = 2 × (5i + 5) = 2 × (5 + 5i).

This expression involves imaginary numbers, so the perimeter cannot be expressed as a real number.

Well explained 👍

FAQ on Square Root of -45

1.What is √(-45) in its simplest form?

The simplest form of √(-45) is 3i√5 because √(-45) can be expressed as √(-1 × 45) = i × √45 = 3i√5.

2.What is the imaginary unit 'i'?

The imaginary unit 'i' is defined as √(-1). It is used to express the square roots of negative numbers, enabling complex number representation.

3.Calculate the square of 3i√5.

The square of 3i√5 is -45. Squaring 3i√5 gives (3i√5)² = 9i² × 5 = 45 × (-1) = -45.

4.Is -45 a real number?

Yes, -45 is a real number. However, its square root is not a real number but an imaginary number.

5.How do you express negative square roots?

Negative square roots are expressed using the imaginary unit 'i'. For example, the square root of -45 is expressed as 3i√5, where 'i' denotes the imaginary component.

Important Glossaries for the Square Root of -45

  • Imaginary unit: The imaginary unit 'i' is defined as √(-1) and is used to express square roots of negative numbers.
     
  • Complex number: A complex number includes both real and imaginary parts, typically expressed in the form a + bi.
     
  • Imaginary number: A number that involves the imaginary unit 'i', such as 3i or 4 + 5i, where 'i' denotes √(-1).
     
  • Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For negative numbers, it involves 'i'.
     
  • Prime factorization: The expression of a number as a product of its prime factors, used to simplify square roots of positive numbers.

What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math

▶

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.