In Exercises 1–20, evaluate each expression, or state that the expression is not a real number. ___ √-36

Verified step by step guidance

1. Identify the expression: \( \sqrt{-36} \).
2. Recognize that the square root of a negative number is not a real number.
3. Recall that the square root of a negative number involves imaginary numbers.
4. Express \( \sqrt{-36} \) using imaginary numbers: \( \sqrt{-36} = \sqrt{36} \times \sqrt{-1} \).
5. Simplify further: \( \sqrt{36} = 6 \) and \( \sqrt{-1} = i \), so \( \sqrt{-36} = 6i \).

Key Concepts

Square Roots

The square root of a number 'x' is a value 'y' such that y² = x. For non-negative numbers, square roots yield real numbers. However, the square root of a negative number does not yield a real number, as no real number squared results in a negative value.

Imaginary Numbers

Imaginary numbers are defined as multiples of the imaginary unit 'i', where i is the square root of -1. This concept allows for the extension of the real number system to include solutions to equations that involve the square roots of negative numbers, such as √-36 = 6i.

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 part. Understanding complex numbers is essential for evaluating expressions involving square roots of negative numbers, as they provide a complete number system.