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Let's simplify the given expression:

\[
\sqrt{12v^{\frac{2}{5}}y^{\frac{8}{7}}} \times \sqrt[3]{3v^{\frac{3}{4}}y^{\frac{4}{9}}}
\]

### Step 1: Rewrite the radicals as fractional exponents.
We know that:

\[
\sqrt{a} = a^{\frac{1}{2}} \quad \text{and} \quad \sqrt[3]{a} = a^{\frac{1}{3}}
\]

Using this, the expression becomes:

\[
(12v^{\frac{2}{5}}y^{\frac{8}{7}})^{\frac{1}{2}} \times (3v^{\frac{3}{4}}y^{\frac{4}{9}})^{\frac{1}{3}}
\]

### Step 2: Apply the power of a power rule.
The rule is \((a^m)^n = a^{m \cdot n}\), so we distribute the exponents:

\[
12^{\frac{1}{2}} v^{\frac{2}{5} \cdot \frac{1}{2}} y^{\frac{8}{7} \cdot \frac{1}{2}} \times 3^{\frac{1}{3}} v^{\frac{3}{4} \cdot \frac{1}{3}} y^{\frac{4}{9} \cdot \frac{1}{3}}
\]

This simplifies to:

\[
12^{\frac{1}{2}} v^{\frac{1}{5}} y^{\frac{4}{7}} \times 3^{\frac{1}{3}} v^{\frac{1}{4}} y^{\frac{4}{27}}
\]

### Step 3: Simplify the constants.
Now simplify the constants \(12^{\frac{1}{2}}\) and \(3^{\frac{1}{3}}\):

\[
12^{\frac{1}{2}} = \sqrt{12} = 2\sqrt{3}
\]
\[
3^{\frac{1}{3}} = \text{cube root of } 3
\]

So, the expression becomes:

\[
2\sqrt{3} \times 3^{\frac{1}{3}} v^{\frac{1}{5}} v^{\frac{1}{4}} y^{\frac{4}{7}} y^{\frac{4}{27}}
\]

### Step 4: Combine the variables.
For \(v\), add the exponents \(\frac{1}{5} + \frac{1}{4}\). To add fractions, find a common denominator:

\[
\frac{1}{5} + \frac{1}{4} = \frac{4}{20} + \frac{5}{20} = \frac{9}{20}
\]

For \(y\), add the exponents \(\frac{4}{7} + \frac{4}{27}\). Find the common denominator:

\[
\frac{4}{7} + \frac{4}{27} = \frac{108}{189} + \frac{28}{189} = \frac{136}{189}
\]

This fraction simplifies to \(\frac{16}{21}\).

### Step 5: Final expression.
Now, putting everything together:

\[
2\sqrt{3} \times 3^{\frac{1}{3}} v^{\frac{9}{20}} y^{\frac{16}{21}}
\]

This is the simplified form of the expression.

Let me know if you need any clarifications, or if you have further questions! Here are some related questions to explore:

1. How do you combine exponents when multiplying terms with the same base?
2. What are the steps for simplifying radicals into fractional exponents?
3. Can you explain how to find a common denominator to add fractions?
4. How do you simplify square roots of constants like \(\sqrt{12}\)?
5. How do you deal with cube roots in an expression?

**Tip:** Always rewrite roots and radicals in terms of exponents when simplifying complex expressions, as it often makes combining terms easier.

Simplify: √(12v^(2/5)y^(8/7)) * ³√(3v^(3/4)y^(4/9))

Learn how to simplify the radical expression √(12v^(2/5)y^(8/7)) * ³√(3v^(3/4)y^(4/9)) step-by-step. This solution covers key concepts such as exponents, radicals, and the power of a power rule. Suitable for high school students in Grades 9-12.

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