In Exercises 29–36, simplify and write the result in standard form.


√−196

Verified step by step guidance

1. Recognize that the expression involves the square root of a negative number, which means it will result in an imaginary number.
2. Recall that the square root of a negative number can be expressed in terms of the imaginary unit \(i\), where \(i = \sqrt{-1}\).
3. Rewrite \(\sqrt{-196}\) as \(\sqrt{196} \times \sqrt{-1}\).
4. Calculate \(\sqrt{196}\), which is a perfect square, resulting in 14.
5. Combine the results to express the simplified form as \(14i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for simplifying expressions involving square roots of negative numbers.

Square Roots of Negative Numbers

The square root of a negative number cannot be expressed as a real number. Instead, it is represented using imaginary numbers. For example, √-196 can be simplified to √196 * √-1, which equals 14i, where 'i' denotes the imaginary unit. This concept is crucial for solving problems that involve square roots of negative values.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. When simplifying expressions involving complex numbers, it is important to express the result in this form for clarity and consistency. For instance, after simplifying √-196, the result should be presented as 0 + 14i, which is equivalent to 14i in standard form.