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 complex numbers, physics, engineering, etc. Here, we will discuss the square root of -225.
The square root is the inverse of the square of a number. For negative numbers, the square root involves imaginary numbers because no real number squared results in a negative product. The square root of -225 is expressed in both radical and exponential form in terms of imaginary numbers. In the radical form, it is expressed as √-225 = 15i, where 'i' is the imaginary unit. In exponential form, it is expressed as (-225)^(1/2) = 15i.
To find the square root of a negative number, we use imaginary numbers. The imaginary unit 'i' is defined as the square root of -1. Thus, for any negative number, its square root can be expressed in terms of 'i'. For example, the square root of -225 can be written as √225 * √(-1), which equals 15i.
The prime factorization method is typically used for positive numbers. Since -225 is negative, we first consider its absolute value, 225. The prime factorization of 225 is 3^2 x 5^2. Therefore, √225 = 15. For -225, we include the imaginary unit: √-225 = √(225) x √(-1) = 15i.
The square root of a negative number requires the use of complex numbers. A complex number has the form a + bi, where 'a' is the real part, and 'bi' is the imaginary part. In this case, since the real part is 0, the square root of -225 is purely imaginary. We calculate it as 15i, where i is the square root of -1.
The square root of negative numbers appears in various applications involving complex numbers. They are used in engineering fields, particularly in electrical engineering and signal processing, where they help in analyzing waveforms and alternating current circuits.
Students often make mistakes when dealing with square roots of negative numbers, such as misapplying the concept of imaginary numbers or forgetting to use 'i'. Let's explore some common errors and how to address them.
If a complex number is given as 0 + √(-225), what is its value?
The value is 15i.
Since √(-225) involves the imaginary unit, the complex number is 0 + 15i, indicating a purely imaginary number with no real part.
Calculate the modulus of the complex number 0 + 15i.
The modulus is 15.
The modulus of a complex number a + bi is √(a^2 + b^2).
Here, a = 0 and b = 15, so the modulus is √(0^2 + 15^2) = 15.
What is the product of (√(-225)) x (√(-225))?
The product is -225.
Multiplying √(-225) by itself gives (-225)^(1/2) x (-225)^(1/2) = -225, since (15i) x (15i) = 225i^2 and i^2 = -1.
Express the square root of -225 in polar form.
In polar form, it is 15(cos(π/2) + isin(π/2)).
The polar form of a complex number a + bi is r(cosθ + isinθ), where r is the modulus, and θ is the argument.
Here, r = 15 and θ = π/2.
If the imaginary part of a complex number is given as the square root of -225, what is the real part?
The real part is 0.
For the complex number 0 + 15i, the real part is 0, indicating no real component.
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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