Square Root of -140
2026-02-21 20:43 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-140

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Last updated on August 5, 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 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

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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.

Problem 1

What is the simplified form of √-140?

Okay, lets begin

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.

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Problem 2

Calculate √-140 * 3.

Okay, lets begin

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.

Well explained 👍

Problem 3

If √-140 is multiplied by itself, what is the result?

Okay, lets begin

The result is -140.

Explanation

(√-140)² = (2√35 * i)² = 4 * 35 * i² = 140 * (-1) = -140, as i² = -1.

Well explained 👍

Problem 4

Find the modulus of √-140.

Okay, lets begin

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.

Well explained 👍

Problem 5

Is the square root of -140 a real number?

Okay, lets begin

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.

Well explained 👍

FAQ on Square Root of -140

1.What is the simplest form of √-140?

The simplest form of √-140 is 2√35 * i.

2.How do you express the square root of a negative number?

The square root of a negative number is expressed in terms of imaginary numbers, using the imaginary unit i.

For example, √-140 = √140 * i.

3.What is the imaginary unit i?

The imaginary unit i is defined as the square root of -1. It is used to express the square roots of negative numbers.

4.Can the square root of a negative number be simplified further?

The square root of a negative number can be simplified by simplifying the square root of its positive part and multiplying by i, but not further into real numbers.

5.What are imaginary numbers used for?

Imaginary numbers are used in complex number theory and have applications in engineering, physics, and applied mathematics, allowing for solutions to equations that do not have real solutions.

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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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.