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 extends into complex numbers when dealing with negative numbers. Here, we will discuss the square root of -2/3.
The square root is the inverse of the square of a number. When dealing with negative numbers, the square root involves imaginary numbers. The square root of -2/3 is expressed using the imaginary unit 'i', where i² = -1. Therefore, the square root of -2/3 can be written as √(-2/3) = √(2/3) * i. Since 2/3 is not a perfect square, √(2/3) is an irrational number, and thus √(-2/3) is a complex number.
To find the square root of a negative fraction, we need to separate the negative sign and use the imaginary unit 'i'. We treat the positive part of the fraction separately from the negative sign. Let us now learn the following methods:
The square root of a negative number is expressed using the imaginary unit 'i'. Let's look at how to express the square root of -2/3:
Step 1: Separate the negative sign and express it with 'i'. √(-2/3) = √(2/3) * i
Step 2: Calculate the square root of the positive fraction. √(2/3) can be written as √2/√3.
Step 3: Simplify the expression. √(-2/3) = (√2/√3) * i
To simplify √(2/3), we can multiply both numerator and denominator by √3 to rationalize the denominator:
Step 1: Multiply by √3/√3: (√2/√3) * (√3/√3) = √6 / 3
Step 2: Express the square root of -2/3 using this result: √(-2/3) = (√6/3) * i
Square roots of negative numbers appear in various fields, especially in electrical engineering and signal processing, where the concept of complex numbers is used extensively. Understanding how to express and manipulate these numbers is critical for solving complex equations and modeling waveforms.
Students often make errors when dealing with square roots of negative numbers, such as forgetting to use the imaginary unit 'i' or incorrectly simplifying fractions. Let's examine some of these mistakes in detail.
How would you express the square root of -4/9 in terms of 'i'?
The expression is (2/3) * i.
The square root of -4/9 can be split into √(-1) * √(4/9). Therefore, it is √4/√9 * i, which simplifies to (2/3) * i.
Calculate the product of √(-2/3) and 3.
The result is √6 * i.
First, express √(-2/3) as (√6/3) * i. Multiply by 3: 3 * (√6/3) * i = √6 * i.
If you have a complex number represented as √(-2/3), what is its magnitude?
The magnitude is √(2/3).
The magnitude of a complex number a * i is |a|. Here, a = √(2/3), so the magnitude is √(2/3).
Explain how to find the square root of -1/4.
The square root is (1/2) * i.
Separate the negative: √(-1/4) = √(-1) * √(1/4). This becomes i * (1/2), or (1/2) * i.
What is the square of the square root of -2/3?
The square is -2/3.
Squaring √(-2/3) gives (-2/3) because (√(-2/3))² = -2/3.
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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