If `x ^((1)/(5)) = y ^((2)/(3)),` then the realtion between x and y is :

Options

A. `x = y ^(2)`
B. `x = y ^(4)`
C. `x ^(4) =y`
D. `x ^(3) = y ^(10)`

Answer

To solve the equation \( x^{\frac{1}{5}} = y^{\frac{2}{3}} \) and find the relation between \( x \) and \( y \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x^{\frac{1}{5}} = y^{\frac{2}{3}} \]

### Step 2: Eliminate the fractions by raising both sides to a common power
To eliminate the fractions, we can raise both sides of the equation to the least common multiple (LCM) of the denominators (5 and 3). The LCM of 5 and 3 is 15. Therefore, we raise both sides to the power of 15:
\[ \left( x^{\frac{1}{5}} \right)^{15} = \left( y^{\frac{2}{3}} \right)^{15} \]

### Step 3: Simplify both sides
Using the power of a power property \((a^{m})^{n} = a^{m \cdot n}\), we simplify both sides:
\[ x^{\frac{15}{5}} = y^{\frac{30}{3}} \]
This simplifies to:
\[ x^{3} = y^{10} \]

### Step 4: Write the final relation
Thus, the relation between \( x \) and \( y \) is:
\[ x^{3} = y^{10} \]

### Step 5: Rearranging the relation (optional)
If needed, we can also express \( y \) in terms of \( x \):
\[ y = x^{\frac{3}{10}} \]

### Final Answer
The relation between \( x \) and \( y \) is:
\[ x^{3} = y^{10} \]

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