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 various fields such as vehicle design, finance, etc. Here, we will discuss the square root of -18.
The square root is the inverse of the square of the number. The number -18 is not a perfect square and does not have a real square root because the square of any real number is non-negative. The square root of -18 is expressed in terms of imaginary numbers. In the radical form, it is expressed as √(-18), whereas in exponential form it is written as (-18)^(1/2). The square root of -18 is an imaginary number and is simplified to 3i√2, where i is the imaginary unit.
The square root of a negative number involves the imaginary unit 'i', which is defined as the square root of -1. When dealing with negative numbers under a square root, the expression can be simplified by factoring out 'i'.
Therefore, the square root of -18 can be simplified by expressing it as √(-1) × √18 = 3i√2.
This method is often used to simplify expressions involving negative numbers under square roots.
To simplify the square root of -18, follow these steps:
Step 1: Recognize that -18 can be written as -1 × 18.
Step 2: Separate the factors under the square root: √(-1 × 18) = √(-1) × √18.
Step 3: Simplify each square root: √(-1) = i and √18 = √(9 × 2) = 3√2.
Step 4: Combine the simplified parts: 3i√2. Thus, the square root of -18 is 3i√2.
When working with imaginary roots, it is essential to understand the properties of the imaginary unit i. The imaginary unit is defined as i = √(-1), and it has the property that i² = -1.
Imaginary roots are often used in complex numbers, which are numbers of the form a + bi, where a and b are real numbers.
Understanding how to manipulate these roots is crucial in fields such as electrical engineering and complex analysis.
Imaginary numbers, like the square root of -18, have practical applications in various scientific and engineering fields. They are used in electrical engineering to analyze AC circuits, where impedance is expressed as a complex number. In control theory, imaginary numbers are used to represent system stability. They also have applications in quantum physics and complex number analysis. Understanding their properties and applications can provide valuable insights into these technical fields.
Students often make mistakes when dealing with imaginary roots, such as misapplying the imaginary unit or incorrectly simplifying expressions. Let us explore some common mistakes and how to avoid them.
What is the value of (√(-18))²?
The value is -18.
When you square the square root of a number, you get the original number back. However, since √(-18) = 3i√2, squaring it gives (3i√2)² = 9i² × 2 = 18 × -1 = -18.
If z = √(-18), what is the modulus of z?
The modulus of z is 3√2.
For a complex number z = a + bi, the modulus is given by |z| = √(a² + b²). Here, z = 0 + 3i√2, so |z| = √(0² + (3√2)²) = √(18) = 3√2.
Find the product of √(-18) and √(-2).
The product is -6i.
First, simplify each square root: √(-18) = 3i√2 and √(-2) = i√2. Multiply these: (3i√2) × (i√2) = 3i² × 2 = 3 × -1 × 2 = -6i.
Calculate the square of the imaginary part of √(-18).
The square is 18.
The imaginary part of √(-18) is 3√2. Squaring this gives (3√2)² = 9 × 2 = 18.
If w = √(-18), express w² in terms of i.
w² = -18.
We have w = 3i√2, so w² = (3i√2)² = 9i² × 2 = 18 × -1 = -18.
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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