question

simplify: $\sqrt4{567n^{11}}$.

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Question

question

simplify: $\sqrt4{567n^{11}}$.

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Simplifying Radicals", "Simplifying Variable Radicals", "Higher Roots" ], "new_concepts": [], "current_concepts": [ "Simplifying Radicals", "Simplifying Variable Radicals", "Higher Roots" ] } ## Factor the radicand into perfect fourth powers $$ \begin{aligned} 567 &= 81 \times 7 = 3^4 \times 7 \ n^{11} &= n^8 \times n^3 = (n^2)^4 \times n^3 \end{aligned} $$ ## Apply the product rule for radicals $$ \begin{aligned} \sqrt[4]{567n^{11}} &= \sqrt[4]{3^4 \cdot n^8 \cdot 7n^3} \ &= \sqrt[4]{3^4 \cdot (n^2)^4} \cdot \sqrt[4]{7n^3} \end{aligned} $$ ## Simplify the perfect fourth powers $$ \sqrt[4]{3^4 \cdot (n^2)^4} \cdot \sqrt[4]{7n^3} = 3n^2\sqrt[4]{7n^3} $$ Simplify: $3n^2\sqrt[4]{7n^3}$. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Simplifying Radicals" ] }

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QUESTION IMAGE

Question

Explanation:

Factor the radicand into perfect fourth powers

Apply the product rule for radicals

Simplify the perfect fourth powers

Answer: