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 concept of square roots, especially of negative numbers, involves complex numbers. Here, we will discuss the square root of -75.
The square root is the inverse of the square of the number. Since -75 is a negative number, its square root involves imaginary numbers. The square root of -75 is expressed in terms of 'i', the imaginary unit, as √(-75) = √(75) × i. The square root of 75 is √(3² × 5) = 5√3. Thus, √(-75) can be expressed as 5√3i, where 'i' is the imaginary unit.
Since -75 is not a perfect square and involves an imaginary number, the typical methods for non-negative numbers are not directly applicable. Instead, we need to consider both the real and imaginary components. Let's explore the concepts involved:
To find the square root of 75, first simplify it using prime factorization:
Step 1: Finding the prime factors of 75
Breaking it down, we get 75 = 3 × 5 × 5 = 3 × 5².
Step 2: Simplify using the prime factors
The square root of 75 is √(3 × 5²) = 5√3.
Since we are dealing with a negative number, we introduce the imaginary unit 'i', where i² = -1.
Step 1: Express -75 as a product of 75 and -1 So, √(-75) = √(75 × -1) = √75 × √(-1).
Step 2: Simplify the expression
Using the imaginary unit, √(-1) = i, thus √(-75) = 5√3i.
Students often make mistakes when dealing with square roots of negative numbers, particularly in using the imaginary unit 'i'. Here are some common errors and tips to avoid them.
If x = √(-75), what is the value of x²?
The value of x² is -75.
Given x = √(-75), we know x can be expressed as 5√3i.
Thus, x² = (5√3i)² = 25 × 3 × i² = 75 × (-1) = -75.
Find the product of √(-75) and √(-3).
The product is √225i², which simplifies to -15.
√(-75) = 5√3i and √(-3) = √3i.
The product is (5√3i) × (√3i) = 5 × 3 × i² = 15 × (-1) = -15.
Calculate the sum of √(-75) and √75.
The sum is 5√3i + 5√3.
√(-75) = 5√3i and √75 = 5√3.
Therefore, the sum is 5√3i + 5√3.
What is the magnitude of the complex number √(-75)?
The magnitude is 15.
The magnitude of a complex number a + bi is √(a² + b²).
Here, a = 0 and b = 5√3, so the magnitude is √(0² + (5√3)²) = √(75) = 15.
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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