find the product with the exponent in simplest form. then, identify the values of $x$ and $y$.

\6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{x}{y}}\

$x = $ , $y = $

Question

find the product with the exponent in simplest form. then, identify the values of $x$ and $y$.

\6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{x}{y}}\

$x = $  , $y = $

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": [ "Exponent Rules", "Algebraic Simplification" ], "new_concepts": [], "current_concepts": [ "Exponent Rules", "Algebraic Simplification", "Rational Exponents" ] } ## Apply the product rule for exponents Using the *Exponent Rules* knowledge point $$ 6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{1}{3} + \frac{1}{4}} $$ ## Find a common denominator and add the fractions Using the *Algebraic Simplification* knowledge point $$ \frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12} $$ ## Equate the exponents to find x and y Using the *Algebraic Simplification* knowledge point $$ \begin{aligned} &6^{\frac{7}{12}} = 6^{\frac{x}{y}}\ &\frac{x}{y} = \frac{7}{12}\ &x = 7,\quad y = 12 \end{aligned} $$ Find the product with the exponent in simplest form. Then, identify the values of $x$ and $y$. $6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{x}{y}}$ $x =$ 7, $y =$ 12 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Exponent Rules" ] }

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QUESTION IMAGE

Question

Explanation:

Apply the product rule for exponents

Find a common denominator and add the fractions

Equate the exponents to find x and y

Answer: