Find each product or quotient. Simplify the answers. √-6 * √-2 / √3

Verified step by step guidance

1. Recognize that the problem involves square roots of negative numbers, which means we will be working with imaginary numbers. Recall that \(\sqrt{-a} = i\sqrt{a}\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
2. Rewrite each square root of a negative number using the imaginary unit \(i\): \(\sqrt{-6} = i\sqrt{6}\) and \(\sqrt{-2} = i\sqrt{2}\). The expression becomes \(\frac{(i\sqrt{6})(i\sqrt{2})}{\sqrt{3}}\).
3. Multiply the numerators: \((i\sqrt{6})(i\sqrt{2}) = i \cdot i \cdot \sqrt{6} \cdot \sqrt{2} = i^2 \sqrt{12}\). Since \(i^2 = -1\), this simplifies to \(-\sqrt{12}\).
4. Simplify \(\sqrt{12}\) by factoring it into \(\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\). So the numerator becomes \(-2\sqrt{3}\).
5. Now the expression is \(\frac{-2\sqrt{3}}{\sqrt{3}}\). Since \(\sqrt{3}\) appears in both numerator and denominator, simplify by canceling \(\sqrt{3}\), leaving \(-2\) as the simplified result.

Key Concepts

Properties of Square Roots

Square roots represent the principal non-negative root of a number. For positive numbers, √a * √b = √(a*b). However, when dealing with negative numbers under the square root, the concept extends to complex numbers, since the square root of a negative number involves imaginary units.

Imaginary Numbers and the Imaginary Unit i

The imaginary unit i is defined as √-1. When taking the square root of negative numbers, express them as √(-1 * positive number) = i√(positive number). This allows simplification of roots of negative numbers into a form involving i, which is essential for working with complex numbers.

Simplifying Expressions with Radicals and Complex Numbers

To simplify expressions involving radicals and complex numbers, first rewrite all radicals with negative radicands using i, then multiply or divide the expressions by combining like terms. Finally, simplify the resulting expression by reducing fractions and combining radicals where possible.