Find each product or quotient. Simplify the answers. √-24 / √8

Verified step by step guidance

1. Recognize that the expression involves square roots of both a negative and a positive number: \(\frac{\sqrt{-24}}{\sqrt{8}}\).
2. Rewrite the square root of the negative number using the imaginary unit \(i\), where \(\sqrt{-1} = i\). So, \(\sqrt{-24} = \sqrt{24} \times i\).
3. Simplify \(\sqrt{24}\) and \(\sqrt{8}\) by factoring out perfect squares: \(\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}\) and \(\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\).
4. Substitute the simplified forms back into the expression: \(\frac{2\sqrt{6} \times i}{2\sqrt{2}}\).
5. Cancel common factors and simplify the fraction \(\frac{\sqrt{6}}{\sqrt{2}}\) by writing it as \(\sqrt{\frac{6}{2}} = \sqrt{3}\), so the expression becomes \(i \times \sqrt{3}\).

Key Concepts

Simplifying Square Roots

Simplifying square roots involves expressing the radicand as a product of perfect squares and other factors, then taking the square root of the perfect squares outside the radical. This process makes the expression easier to work with and often reveals simpler forms.

Division of Radicals

When dividing square roots, the quotient of the radicals can be expressed as the square root of the quotient of their radicands, i.e., √a / √b = √(a/b), provided b ≠ 0. This property helps combine or simplify expressions involving radicals.

Complex Numbers and Imaginary Unit

The square root of a negative number involves the imaginary unit i, where i² = -1. For example, √-24 can be rewritten as √(24) * i. Understanding this allows simplification of roots with negative radicands by separating the imaginary part.