To find the product [tex]\( 6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} \)[/tex] and express it in the form [tex]\( 6^{\frac{x}{y}} \)[/tex], we follow these steps:
1. Combine the exponents:
When you multiply two exponents with the same base, you add the exponents:
[tex]\[
6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\left(\frac{1}{3} + \frac{1}{4}\right)}
\][/tex]
2. Find a common denominator:
The fractions [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex] need to have a common denominator to be added. The least common multiple (LCM) of 3 and 4 is 12.
3. Convert the fractions:
Rewrite [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex] with a denominator of 12:
[tex]\[
\frac{1}{3} = \frac{4}{12} \quad \text{and} \quad \frac{1}{4} = \frac{3}{12}
\][/tex]
4. Add the fractions:
Now, add the numerators while keeping the common denominator:
[tex]\[
\frac{4}{12} + \frac{3}{12} = \frac{4 + 3}{12} = \frac{7}{12}
\][/tex]
5. Rewriting the expression:
So, the product [tex]\( 6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} \)[/tex] can be written as:
[tex]\[
6^{\left(\frac{1}{3} + \frac{1}{4}\right)} = 6^{\frac{7}{12}}
\][/tex]
Thus, we have:
[tex]\[
6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{7}{12}}
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are:
[tex]\[
x = 7, \quad y = 12
\][/tex]
