Find each product or quotient. Simplify the answers. √-10 / √-40

검증된 단계별 안내

Recognize that the problem involves the quotient of two square roots with negative radicands: \(\frac{\sqrt{-10}}{\sqrt{-40}}\).

Rewrite each square root of a negative number using the imaginary unit \(i\), where \(i = \sqrt{-1}\). So, \(\sqrt{-10} = \sqrt{10} \cdot i\) and \(\sqrt{-40} = \sqrt{40} \cdot i\).

Substitute these into the expression: \(\frac{\sqrt{10} \cdot i}{\sqrt{40} \cdot i}\).

Cancel the common factor \(i\) in numerator and denominator, leaving \(\frac{\sqrt{10}}{\sqrt{40}}\).

Simplify the quotient of square roots by writing it as \(\sqrt{\frac{10}{40}}\) and then simplify the fraction inside the square root before simplifying the radical.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Simplifying Square Roots of Negative Numbers

Square roots of negative numbers involve imaginary numbers, since the square root of a negative value is not real. This is expressed using the imaginary unit 'i', where i² = -1. For example, √-10 can be rewritten as √10 × i.

Properties of Radicals in Division

When dividing square roots, the quotient rule applies: √a / √b = √(a/b), provided a and b are non-negative. This property helps simplify expressions by combining radicals under a single root before further simplification.

Simplifying Radicals by Factoring

Simplifying radicals often involves factoring the number inside the root into perfect squares and other factors. Extracting the square root of perfect squares simplifies the expression, such as √40 = √(4×10) = 2√10.