If x, y are two positive real number and ` x^(1//3) = y^(1//4)` , then which of the following relations is true ?

Options

A. `x^(3) = y^(4)`
B. `x^(3) = y`
C. ` x = y^(4)`
D. `x^(20) = y^(15)`

Answer

To solve the problem where \( x^{1/3} = y^{1/4} \) and we need to find the correct relation among the given options, we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x^{1/3} = y^{1/4} \]

### Step 2: Eliminate the roots by raising both sides to a common power
To eliminate the fractional exponents, we can raise both sides to the least common multiple (LCM) of the denominators of the exponents. The LCM of 3 and 4 is 12. Thus, we raise both sides to the power of 12:
\[ (x^{1/3})^{12} = (y^{1/4})^{12} \]

### Step 3: Simplify both sides
Using the property of exponents \((a^{m})^{n} = a^{m \cdot n}\), we can simplify:
\[ x^{12/3} = y^{12/4} \]
This simplifies to:
\[ x^4 = y^3 \]

### Step 4: Rearranging the equation
Now, we can express \(x\) in terms of \(y\) or vice versa. We can rewrite the equation as:
\[ x^4 = y^3 \]

### Step 5: Raise both sides to find a relation
To find a relation that matches one of the options, we can raise both sides to the power of 5:
\[ (x^4)^5 = (y^3)^5 \]
This simplifies to:
\[ x^{20} = y^{15} \]

### Conclusion
Thus, we have derived the relation:
\[ x^{20} = y^{15} \]
This matches option 4.

### Final Answer
The correct relation is:
\[ \text{Option 4: } x^{20} = y^{15} \]

---