Given that `root(3) (3^x) = 5^(1//4) and root(4)(5^y) = sqrt(3)`, then find the value of `2xy`.

Answer

To solve the problem step by step, we start with the given equations:

1. \( \sqrt[3]{3^x} = 5^{1/4} \)
2. \( \sqrt[4]{5^y} = \sqrt{3} \)

### Step 1: Rewrite the equations in exponential form

The first equation can be rewritten using the property of exponents:

\[ 3^{x/3} = 5^{1/4} \]

The second equation can also be rewritten:

\[ 5^{y/4} = 3^{1/2} \]

### Step 2: Eliminate the roots by raising both sides to the appropriate powers

For the first equation, we can cube both sides to eliminate the cube root:

\[ 3^x = (5^{1/4})^3 \]
\[ 3^x = 5^{3/4} \]

For the second equation, we can raise both sides to the power of 4 to eliminate the fourth root:

\[ 5^y = (\sqrt{3})^4 \]
\[ 5^y = 3^2 \]

### Step 3: Set up the equations for comparison

Now we have two equations:

1. \( 3^x = 5^{3/4} \) (Equation 1)
2. \( 5^y = 3^2 \) (Equation 2)

### Step 4: Express \(x\) and \(y\) in terms of logarithms

From Equation 1, we can express \(x\):

Taking logarithm base 3 on both sides:

\[ x = \log_3(5^{3/4}) = \frac{3}{4} \log_3(5) \]

From Equation 2, we can express \(y\):

Taking logarithm base 5 on both sides:

\[ y = \log_5(3^2) = 2 \log_5(3) \]

### Step 5: Calculate \(2xy\)

Now we need to find \(2xy\):

\[ 2xy = 2 \left(\frac{3}{4} \log_3(5)\right) \left(2 \log_5(3)\right) \]

This simplifies to:

\[ 2xy = 3 \cdot \frac{3}{2} \cdot \log_3(5) \cdot \log_5(3) \]

Using the change of base formula, we know that:

\[ \log_3(5) \cdot \log_5(3) = 1 \]

Thus:

\[ 2xy = 3 \]

### Final Answer:

The value of \(2xy\) is \(3\).

---