Questions: √[12]y^5 · √[6]y^5 = y^(5/6 + 5/12)

Solution Steps

To solve the given problem, we need to convert the radicals into exponential form and then apply the rules of exponents to simplify the expression. Specifically, we will use the property that \(\sqrt[n]{a^m} = a^{m/n}\). After converting, we will combine the exponents by adding them together.

1. Convert each radical to its exponential form.
2. Add the exponents together.
3. Simplify the resulting expression.

We start with the expression: \[ \sqrt[12]{y^{5}} \cdot \sqrt[6]{y^{5}} \] This can be rewritten using exponential notation: \[ y^{\frac{5}{12}} \cdot y^{\frac{5}{6}} \]

Next, we add the exponents: \[ \frac{5}{12} + \frac{5}{6} \] To add these fractions, we need a common denominator. The least common multiple of 12 and 6 is 12. Thus, we convert \(\frac{5}{6}\) to \(\frac{10}{12}\): \[ \frac{5}{12} + \frac{10}{12} = \frac{15}{12} \]

Now we simplify \(\frac{15}{12}\): \[ \frac{15}{12} = \frac{5}{4} \] Thus, the expression simplifies to: \[ y^{\frac{5}{4}} \]

Final Answer