Question

find the product with the exponent in simplest form. then, identify the values of x and y.
$6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{x}{y}}$

Explanation:

Step1: Apply exponent product rule

When multiplying exponents with the same base, we add the exponents: \( a^m \cdot a^n = a^{m + n} \). So for \( 6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} \), we add \( \frac{1}{3} \) and \( \frac{1}{4} \).

Step2: Add the fractions

To add \( \frac{1}{3} + \frac{1}{4} \), find a common denominator, which is 12. Convert the fractions: \( \frac{1}{3} = \frac{4}{12} \) and \( \frac{1}{4} = \frac{3}{12} \). Then add: \( \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \). So \( 6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{7}{12}} \), which means \( \frac{x}{y} = \frac{7}{12} \), so \( x = 7 \) and \( y = 12 \).

Answer:

\( x = 7 \), \( y = 12 \)