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To simplify the given expression into a single radical expression, let’s analyze the expression step-by-step.

The problem shows:
\[
\sqrt[3]{y} \cdot \sqrt[5]{y^2}
\]

### Step 1: Convert Radicals to Rational Exponents
Rewrite each radical as a power of \( y \):
1. \(\sqrt[3]{y}\) becomes \( y^{\frac{1}{3}} \).
2. \(\sqrt[5]{y^2}\) becomes \( y^{\frac{2}{5}} \).

So the expression is now:
\[
y^{\frac{1}{3}} \cdot y^{\frac{2}{5}}
\]

### Step 2: Apply the Product of Powers Property
When multiplying powers of the same base, you add the exponents:
\[
y^{\frac{1}{3} + \frac{2}{5}}
\]

### Step 3: Find a Common Denominator
To add \(\frac{1}{3}\) and \(\frac{2}{5}\), convert them to a common denominator. The least common multiple of 3 and 5 is 15:
1. \(\frac{1}{3} = \frac{5}{15}\)
2. \(\frac{2}{5} = \frac{6}{15}\)

Now we can add the exponents:
\[
\frac{5}{15} + \frac{6}{15} = \frac{11}{15}
\]

### Step 4: Rewrite the Expression
The expression simplifies to:
\[
y^{\frac{11}{15}}
\]

### Final Answer in Radical Form
Since \( y^{\frac{11}{15}} \) is equivalent to the radical expression:
\[
\sqrt[15]{y^{11}}
\]

Thus, the answer is:
\[
\sqrt[15]{y^{11}}
\]

Would you like a further breakdown or have any questions?

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**Related Questions:**
1. How do you convert between radical and rational exponent forms?
2. What is the rule for multiplying terms with exponents?
3. How do you find a common denominator when adding fractions?
4. What if the exponents had a negative sign—how would that affect the solution?
5. Can the process be applied to other variables or expressions?

**Tip:** When adding fractions, always find the least common multiple to make the calculation simpler.

Use rational exponents to write as a single radical expression. Assume that all variables represent positive real numbers. 3√y ⋅ 5√y^2

Learn how to simplify the expression 3√y ⋅ 5√y^2 by converting radicals to rational exponents and combining them into a single radical expression. This problem covers key algebraic concepts such as rational exponents, properties of exponents, and radicals. Suitable for students in Grades 9-10.

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