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

Verified step by step guidance

1. Recognize that the expression involves the square root of a negative number, which means we will be dealing with imaginary numbers.
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 $\sqrt{-49}$ as $\sqrt{49}\times \sqrt{-1}$.
4. Calculate $\sqrt{49}$, which is 7, and multiply it by $\sqrt{-1}$, which is $i$.
5. Express the result in standard form as $7i$.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'i' is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for simplifying expressions involving square roots of negative numbers, such as √-49.

Imaginary Unit (i)

The imaginary unit 'i' is defined as i = √-1. It allows for the extension of the real number system to include solutions to equations that do not have real solutions, such as the square root of negative numbers. In the case of √-49, it can be rewritten as 7i, where 7 is the square root of 49.

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 coefficient of the imaginary part. When simplifying expressions like √-49, it is important to express the result in this form to clearly identify the real and imaginary components, resulting in 0 + 7i.