perform the indicated operation & simplify. express the answer in terms of $i$ (as a complex number).

$\sqrt{-4} \cdot \sqrt{-16} =$

Question

perform the indicated operation & simplify. express the answer in terms of $i$ (as a complex number).

$\sqrt{-4} \cdot \sqrt{-16} =$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Imaginary Unit", "Complex Numbers", "Simplifying Radicals" ], "new_concepts": [], "current_concepts": [ "Imaginary Unit", "Complex Numbers", "Simplifying Radicals" ] } ## Express radicals using the imaginary unit $$ \begin{aligned} \sqrt{-4} &= \sqrt{4} \cdot \sqrt{-1} = 2i \ \sqrt{-16} &= \sqrt{16} \cdot \sqrt{-1} = 4i \end{aligned} $$ ## Multiply the simplified imaginary numbers $$ (2i) \cdot (4i) = 8i^2 $$ ## Simplify using the definition of i squared $$ \begin{aligned} i^2 &= -1 \ 8i^2 &= 8(-1) = -8 \end{aligned} $$ Perform the indicated operation & simplify. Express the answer in terms of $i$ (as a complex number). $\sqrt{-4} \cdot \sqrt{-16} =$ -8 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Complex Numbers" ] }

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QUESTION IMAGE

Question

Explanation:

Express radicals using the imaginary unit

Multiply the simplified imaginary numbers

Simplify using the definition of i squared

Answer: