111 LearnersLast updated on May 26th, 2025
Square Root of -140
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 is used in various fields, including complex number theory. Here, we will discuss the square root of -140.
What is the Square Root of -140?
The square root is the inverse of the square of the number. Since -140 is a negative number, it does not have a real square root. Instead, its square root is expressed in terms of imaginary numbers. In standard form, the square root of -140 is written as √-140 = √140 * i, where i is the imaginary unit, defined as √-1.
Understanding the Square Root of -140
To understand the square root of -140, we need to consider the concept of imaginary numbers. Imaginary numbers are used when dealing with the square roots of negative numbers. The square root of -140 can be expressed as √140 * i. Let's explore the methods to express the square root of -140 in a simplified form:
1. Simplifying the square root of the positive part: √140
2. Multiplying by the imaginary unit i
Simplifying √140
We can simplify √140 by using prime factorization:
Step 1: Prime factorization of 140 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7
Step 2: Pairing the prime factors From the factors, we can take one pair of 2 outside the square root: √140 = √(2² × 5 × 7) = 2√(5 × 7) = 2√35
Expressing the Square Root of -140
Now, let's express the square root of -140 using the simplified form of √140 and the imaginary unit i: √-140 = √140 * i = 2√35 * i
Common Mistakes and How to Avoid Them in the Square Root of -140
When finding the square root of a negative number, students often make mistakes due to misunderstanding the concept of imaginary numbers. Let's look at a few common mistakes and how to avoid them.
Mistake 1
Forgetting the Imaginary Component
Students may forget to include the imaginary unit i when finding the square root of a negative number. Remember, the square root of a negative number must include i.
For instance, √-140 should be written as 2√35 * i, not just 2√35.
Mistake 2
Confusing Real and Imaginary Roots
It's important to differentiate between real and imaginary roots. While a real number has two real roots (positive and negative), a negative number has no real roots but two imaginary roots.
For example, √-140 = ±2√35 * i.
Mistake 3
Misapplying Prime Factorization
When simplifying the square root of the positive part of a negative number, students may incorrectly factor the number. Ensure to correctly perform the prime factorization to simplify expressions like √140.
Mistake 4
Confusing the Square Root Symbol with Other Roots
Distinguishing between square roots and cube roots (or other roots) is crucial. For example, the square root of -140 is expressed with an imaginary unit, while cube roots do not involve imaginary numbers in the same context.
Mistake 5
Ignoring the Properties of i
The imaginary unit i has specific properties, such as i² = -1.
Ignoring these properties can lead to incorrect conclusions.
Always apply the correct properties when working with i.
Square Root of -140 Examples
Problem 1
What is the simplified form of √-140?
The simplified form is 2√35 * i.
Explanation
To simplify √-140, we first simplify √140 using prime factorization to get 2√35, then multiply by the imaginary unit i to account for the negative sign, resulting in 2√35 * i.
Problem 2
Calculate √-140 * 3.
The result is 6√35 * i.
Explanation
First, find the square root of -140, which is 2√35 * i.
Multiply this by 3: 2√35 * i * 3 = 6√35 * i.
Problem 3
If √-140 is multiplied by itself, what is the result?
The result is -140.
Explanation
(√-140)² = (2√35 * i)² = 4 * 35 * i² = 140 * (-1) = -140, as i² = -1.
Problem 4
Find the modulus of √-140.
The modulus is 2√35.
Explanation
The modulus of a complex number a + bi is √(a² + b²).
For √-140 = 0 + 2√35i, the modulus is √(0² + (2√35)²) = 2√35.
Problem 5
Is the square root of -140 a real number?
No, it is not a real number.
Explanation
The square root of a negative number is not real; it involves the imaginary unit i.
Thus, √-140 = 2√35 * i is not a real number.
FAQ on Square Root of -140
1.What is the simplest form of √-140?
2.How do you express the square root of a negative number?
3.What is the imaginary unit i?
4.Can the square root of a negative number be simplified further?
5.What are imaginary numbers used for?
6.How does learning Algebra help students in Oman make better decisions in daily life?
7.How can cultural or local activities in Oman support learning Algebra topics such as Square Root of -140?
8.How do technology and digital tools in Oman support learning Algebra and Square Root of -140?
9.Does learning Algebra support future career opportunities for students in Oman?
Important Glossaries for the Square Root of -140
- Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For negative numbers, it involves imaginary numbers.
- Imaginary number: An imaginary number is a number that can be written as a real number multiplied by the imaginary unit i, where i is the square root of -1.
- Complex number: A complex number is a number that has both a real part and an imaginary part, expressed in the form a + bi.
- Prime factorization: Breaking down a number into its basic building blocks, which are its prime factors.
- Modulus of a complex number: The modulus is the distance of the complex number from the origin in the complex plane, calculated as √(a² + b²) for a number a + bi.
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About BrightChamps in Oman
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.
