Find each product or quotient. Simplify the answers. √-3 * √-8

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Recognize that the square root of a negative number involves imaginary numbers. Recall that \(\sqrt{-a} = \sqrt{a} \cdot i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).

Rewrite each square root separately using the imaginary unit: \(\sqrt{-3} = \sqrt{3} \cdot i\) and \(\sqrt{-8} = \sqrt{8} \cdot i\).

Express the product \(\sqrt{-3} \times \sqrt{-8}\) as \((\sqrt{3} \cdot i) \times (\sqrt{8} \cdot i)\).

Multiply the terms: combine the square roots and the imaginary units separately, resulting in \(\sqrt{3} \times \sqrt{8} \times i \times i\).

Simplify \(\sqrt{3} \times \sqrt{8}\) to \(\sqrt{24}\), and since \(i \times i = i^2 = -1\), rewrite the expression as \(\sqrt{24} \times (-1)\). Then simplify \(\sqrt{24}\) by factoring out perfect squares.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Square Roots of Negative Numbers

The square root of a negative number is not a real number; it involves imaginary numbers. Specifically, √-a = i√a, where i is the imaginary unit defined by i² = -1. This allows us to work with roots of negative numbers in the complex number system.

Multiplication of Square Roots

When multiplying square roots, the property √a * √b = √(a*b) holds for non-negative numbers. For complex numbers, this property extends by considering the imaginary unit i, allowing simplification of products involving roots of negative numbers.

Simplifying Expressions with Imaginary Numbers

Simplifying expressions with imaginary numbers involves combining like terms and reducing radicals. After expressing roots of negative numbers using i, multiply or divide the terms, then simplify the radical and coefficients to write the answer in standard form a + bi.

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