In Exercises 29–36, simplify and write the result in standard form. √-49

Key Concepts

Imaginary Numbers

Imaginary numbers are defined as multiples of the imaginary unit 'i', where i is the square root of -1. They arise when taking the square root of negative numbers, which do not have real solutions. For example, √-1 = i, and thus √-49 can be expressed as √(49) * √(-1) = 7i.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' is the real part and 'b' is the imaginary part. In the context of simplifying expressions involving square roots of negative numbers, it is essential to represent the result in this format. For √-49, the standard form would be 0 + 7i, simplifying to just 7i.

Properties of Square Roots

The properties of square roots state that √(a * b) = √a * √b and √(a/b) = √a / √b, applicable for non-negative 'a' and 'b'. When dealing with negative numbers, it is crucial to separate the real and imaginary components. This property allows us to break down √-49 into √49 and √-1, facilitating the simplification process.