Simplify and write the result in standard form. √-108

Verified step by step guidance

1. Recognize that the square root of a negative number involves the imaginary unit 'i', where i = √-1. Rewrite √-108 as √(-1 × 108).
2. Use the property of square roots that √(a × b) = √a × √b to separate the square root: √(-1 × 108) = √-1 × √108.
3. Replace √-1 with 'i', giving i × √108.
4. Simplify √108 by factoring 108 into its prime factors. Notice that 108 = 36 × 3, and since 36 is a perfect square, √108 = √36 × √3 = 6√3.
5. Combine the results to write the expression in standard form: i × 6√3, or equivalently 6i√3.

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

When taking the square root of a negative number, the result involves the imaginary unit 'i'. For example, √-1 equals 'i', and √-n can be expressed as i√n. This concept is crucial for simplifying expressions like √-108, as it allows us to rewrite the expression in terms of real and imaginary components.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. To express a complex number in standard form, one must separate the real and imaginary parts after simplification. This is important for clarity and consistency in mathematical communication, especially when dealing with complex numbers derived from square roots of negative values.