In Exercises 29–36, simplify and write the result in standard form. √-108

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. Rewrite \( \sqrt{-108} \) as \( \sqrt{-1 \times 108} \).
3. Use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to separate the square root: \( \sqrt{-1} \times \sqrt{108} \).
4. Recall that \( \sqrt{-1} = i \), where \( i \) is the imaginary unit.
5. Simplify \( \sqrt{108} \) by finding its prime factorization: \( 108 = 2^2 \times 3^3 \), and then simplify further to express the result in terms of \( i \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, typically 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.

Imaginary Unit (i)

The imaginary unit 'i' is defined as the square root of -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 context of the question, √-108 can be rewritten using 'i' to facilitate simplification.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. When simplifying expressions involving square roots of negative numbers, it is important to express the result in this form to clearly identify the real and imaginary components. For example, simplifying √-108 involves breaking it down into its real and imaginary parts to achieve the standard form.