If the square root of x is the cube root of y, then the relation between x and y is

Options

A. `x^(3) = y^(2)`
B. `x^(2) = y^(3)`
C. x = y
D. `x^(6) = y^(5)`

Answer

To solve the problem, we need to establish the relationship between \( x \) and \( y \) given that the square root of \( x \) is equal to the cube root of \( y \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \sqrt{x} = \sqrt[3]{y} \]

2. **Rewrite the square root and cube root in exponent form**:
\[ x^{1/2} = y^{1/3} \]

3. **To eliminate the fractions in the exponents, find a common multiple**:
The least common multiple of 2 and 3 is 6. Therefore, we will raise both sides of the equation to the power of 6:
\[ (x^{1/2})^6 = (y^{1/3})^6 \]

4. **Simplify both sides**:
\[ x^{6 \cdot \frac{1}{2}} = y^{6 \cdot \frac{1}{3}} \]
This simplifies to:
\[ x^3 = y^2 \]

5. **Express the relationship between \( x \) and \( y \)**:
The relationship can be written as:
\[ x^3 = y^2 \]
or alternatively,
\[ y = x^{3/2} \]

### Final Relation:
The relation between \( x \) and \( y \) is:
\[ x^3 = y^2 \]