Perform the indicated operations and write the result in standard form. $\sqrt{-8} \left( \sqrt{-3} - \sqrt{-5} \right)$

Verified step by step guidance

1. Recognize that the expression involves square roots of negative numbers, which means we will be working with imaginary numbers. Recall that \( \sqrt{-1} = i \).
2. Rewrite the square roots of negative numbers using \( i \): \( \sqrt{-8} = \sqrt{8} \cdot i \) and \( \sqrt{-3} = \sqrt{3} \cdot i \).
3. Simplify \( \sqrt{8} \) as \( 2\sqrt{2} \), so \( \sqrt{-8} = 2\sqrt{2}i \).
4. Substitute the expressions back into the original problem: \( (2\sqrt{2}i)(\sqrt{3}i - \sqrt{5}) \).
5. Distribute \( 2\sqrt{2}i \) across the terms inside the parentheses: \( 2\sqrt{2}i \cdot \sqrt{3}i - 2\sqrt{2}i \cdot \sqrt{5} \).

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 √-1. Understanding complex numbers is essential for performing operations involving square roots of negative numbers, as they allow us to extend the real number system.

Square Roots of Negative Numbers

The square root of a negative number is not defined within the real number system, but it can be expressed using imaginary numbers. For example, √-8 can be simplified to 2√2i, where 'i' represents the imaginary unit. This concept is crucial for solving problems that involve square roots of negative values, as it allows for the manipulation of these expressions in algebraic operations.

Standard Form of Complex Numbers

The standard form of a complex number is typically written as a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, such as addition or multiplication, it is important to express the final result in this standard form for clarity and consistency. This involves combining like terms and ensuring that the imaginary unit 'i' is properly accounted for in the final expression.