112 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.
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.
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.
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)?
5.How does learning Algebra help students in Vietnam make better decisions in daily life?
6.How can cultural or local activities in Vietnam support learning Algebra topics such as Square Root of -75?
7.How do technology and digital tools in Vietnam support learning Algebra and Square Root of -75?
8.Does learning Algebra support future career opportunities for students in Vietnam?
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.
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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.
