what is the simplified square root of -20?
The correct answer and explanation is:
Correct Answer:
The simplified square root of -20 is 2√5i
Explanation:
To simplify the square root of a negative number, it is important to understand the concept of imaginary numbers. The square root of any negative number is not a real number because no real number squared gives a negative result. However, in mathematics, a special unit called i is used to represent the square root of -1. That is:
√(-1) = i
So, to simplify √(-20), begin by separating the negative sign from the number:
√(-20) = √(-1 × 20)
Using the property of square roots:
√(-1 × 20) = √(-1) × √(20) = i × √(20)
Next, simplify √(20). The number 20 can be broken down into its prime factors:
20 = 4 × 5
Since 4 is a perfect square:
√(20) = √(4 × 5) = √(4) × √(5) = 2√5
Now put everything together:
i × √(20) = i × 2√5 = 2√5i
This result is in simplest form. The expression 2√5i is called an imaginary number because it involves the imaginary unit i. While it is not a real number, it is still a valid and useful quantity in mathematics, especially in fields such as engineering, physics, and complex analysis.
Imaginary numbers are often used when solving equations that have no real solutions, such as quadratic equations where the discriminant is negative. In such cases, square roots of negative numbers naturally appear, and the concept of i allows us to continue solving these problems meaningfully.