Last updated on May 26th, 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.
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 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
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.
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.
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.
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.
Can you help Alex find the imaginary square root of -81?
The imaginary square root is 9i.
The square root of -81 involves the imaginary unit 'i'.
First, express -81 as -1 × 81. √(-81) = √(-1 × 81) = √(-1) × √81 = i × 9 = 9i.
A circuit has a negative impedance of -64 ohms. What is the imaginary square root value?
8i ohms
To find the square root of the negative impedance: √(-64) = √(-1 × 64) = √(-1) × √64 = i × 8 = 8i.
Calculate 5 times the square root of -36.
30i
First, find the square root of -36: √(-36) = √(-1 × 36) = i × 6 = 6i.
Next, multiply by 5: 5 × 6i = 30i.
What will be the imaginary square root of (-16 + 4)?
The imaginary square root is 4i.
Calculate the sum: -16 + 4 = -12.
Find the square root: √(-12) = √(-1 × 12) = i × √12 = i × 2√3 = 2i√3.
Find the perimeter of a rectangle if its length ‘l’ is √(-25) units and the width ‘w’ is 5 units.
The perimeter of the rectangle is not a real number.
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.
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.