103 LearnersLast updated on May 26th, 2025
Square Root of -75
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.
What is 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.
Struggling with Math?
Get 1:1 Coaching to Boost Grades Fast !
Finding the Square Root of -75
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:
- Imaginary unit
- Simplifying the square root of positive numbers
- Combining results using the imaginary unit
Square Root of 75 by Simplification
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.
Square Root of -75 Using Imaginary Numbers
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.
Common Mistakes and How to Avoid Them in the Square Root of -75
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.
Mistake 1
Ignoring the Imaginary Unit
Students may forget that the square root of a negative number involves the imaginary unit 'i'. Always remember to include 'i' in expressions involving square roots of negative numbers.
For example, √(-75) should be written as 5√3i, not just 5√3.
Mistake 2
Incorrect Simplification of Square Roots
When simplifying square roots, ensure you factor the number correctly.
For example, √75 should be simplified to 5√3, not just √75. Always check for perfect squares within the factorization.
Mistake 3
Confusing Imaginary and Real Components
Students might mix up real and imaginary parts. Keep real components separate from imaginary ones. In √(-75), the real part is √75, and the imaginary part is 'i'.
Mistake 4
Misinterpretation of the Imaginary Unit
Students might not clearly understand that 'i' represents √(-1). Reinforce that i² = -1 and that 'i' is crucial for expressing square roots of negative numbers.
Level Up with a Math Certification!
2X Faster Learning (Grades 1-12)
Square Root of -75 Examples
Problem 1
If x = √(-75), what is the value of x²?
The value of x² is -75.
Explanation
Given x = √(-75), we know x can be expressed as 5√3i.
Thus, x² = (5√3i)² = 25 × 3 × i² = 75 × (-1) = -75.
Problem 2
Find the product of √(-75) and √(-3).
The product is √225i², which simplifies to -15.
Explanation
√(-75) = 5√3i and √(-3) = √3i.
The product is (5√3i) × (√3i) = 5 × 3 × i² = 15 × (-1) = -15.
Problem 3
Calculate the sum of √(-75) and √75.
The sum is 5√3i + 5√3.
Explanation
√(-75) = 5√3i and √75 = 5√3.
Therefore, the sum is 5√3i + 5√3.
Problem 4
What is the magnitude of the complex number √(-75)?
The magnitude is 15.
Explanation
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.
Turn your child into a math star!
#1 Math Hack Schools Won't Teach!
FAQ on Square Root of -75
1.What is the simplest form of √(-75)?
2.Why does √(-75) involve an imaginary number?
3.Is -75 a perfect square?
4.What is the square of √(-75)?
Struggling with Math?
Get 1:1 Coaching to Boost Grades Fast !
Important Glossaries for the Square Root of -75
- Imaginary Unit: The imaginary unit 'i' is defined as √(-1). It is used to express square roots of negative numbers.
- Complex Number: A number that has both a real part and an imaginary part, typically expressed in the form a + bi.
- Prime Factorization: Breaking down a number into its basic prime factors. For example, 75 = 3 × 5 × 5.
- Square Root: The value that, when multiplied by itself, gives the original number. For negative numbers, it involves imaginary numbers.
- Magnitude: The absolute value or length of a complex number, calculated as √(a² + b²) for a complex number a + bi.
Explore More algebra
![Important Math Links Icon]()
Previous to Square Root of -75
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.
