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 is used in various fields, including complex number theory. Here, we will discuss 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.
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
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
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
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.
What is the simplified form of √-140?
The simplified form is 2√35 * i.
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.
Calculate √-140 * 3.
The result is 6√35 * i.
First, find the square root of -140, which is 2√35 * i.
Multiply this by 3: 2√35 * i * 3 = 6√35 * i.
If √-140 is multiplied by itself, what is the result?
The result is -140.
(√-140)² = (2√35 * i)² = 4 * 35 * i² = 140 * (-1) = -140, as i² = -1.
Find the modulus of √-140.
The modulus is 2√35.
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.
Is the square root of -140 a real number?
No, it is not a real number.
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.
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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