solve the equation. write your answers in exact, simplified form.

\6z^5 = -384z^2\

the solution set is $\{\quad\}$.

Question

solve the equation. write your answers in exact, simplified form.

\6z^5 = -384z^2\

the solution set is $\{\quad\}$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Fractional Equations", "Greatest Common Factor", "Sum of Cubes", "Zero Product Property", "Imaginary Numbers" ], "new_concepts": [ "Negative Exponent Equations" ], "current_concepts": [ "Fractional Equations", "Greatest Common Factor", "Sum of Cubes", "Zero Product Property", "Imaginary Numbers", "Negative Exponent Equations" ] } ## Clear negative exponents Using the *Fractional Equations* knowledge point $$ \begin{aligned} &6z^5 = -384z^{-2}, \quad z eq 0 \ &6z^5 = -\frac{384}{z^2} \ &6z^7 = -384 \end{aligned} $$ ## Isolate the variable power Using the *Greatest Common Factor* knowledge point $$ \begin{aligned} &z^7 = \frac{-384}{6} \ &z^7 = -64 \end{aligned} $$ ## Solve for z Using the *Zero Product Property* knowledge point $$ \begin{aligned} &z = \sqrt[7]{-64} \ &z = -2^{6/7} \end{aligned} $$ Solve the equation. Write your answers in exact, simplified form. $6z^5 = -384z^{-2}$ The solution set is $\{-\sqrt[7]{64}\}$. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Fractional Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Clear negative exponents

Isolate the variable power

Solve for z

Answer: