x-y relation is given by `y=x^(n)` calculate the value of n in the following :-
(i)`y=(x^(2))/(x^(2//3))`
(ii)`y=(x^(3//2))/(x^(1))`
(iii)`y=(x^(1//2))/(x^(1//2))`
(iv)`y=(x^(3//2))/(sqrt(x))`

Answer

To solve the given problem, we need to find the value of \( n \) in the relation \( y = x^n \) for each of the provided equations. We will use the property of exponents that states \( \frac{x^a}{x^b} = x^{a-b} \).

### Step-by-Step Solution:

#### (i) \( y = \frac{x^2}{x^{2/3}} \)

1. **Identify the exponents**: Here, \( a = 2 \) and \( b = \frac{2}{3} \).
2. **Apply the exponent rule**:
\[ y = x^{2 - \frac{2}{3}} \]
3. **Find a common denominator**: The common denominator for 2 and \(\frac{2}{3}\) is 3.
\[ 2 = \frac{6}{3} \]
4. **Subtract the exponents**:
\[ y = x^{\frac{6}{3} - \frac{2}{3}} = x^{\frac{4}{3}} \]
5. **Conclusion**: Therefore, \( n = \frac{4}{3} \).

#### (ii) \( y = \frac{x^{3/2}}{x^1} \)

1. **Identify the exponents**: Here, \( a = \frac{3}{2} \) and \( b = 1 \).
2. **Apply the exponent rule**:
\[ y = x^{\frac{3}{2} - 1} \]
3. **Convert 1 to a fraction**:
\[ 1 = \frac{2}{2} \]
4. **Subtract the exponents**:
\[ y = x^{\frac{3}{2} - \frac{2}{2}} = x^{\frac{1}{2}} \]
5. **Conclusion**: Therefore, \( n = \frac{1}{2} \).

#### (iii) \( y = \frac{x^{1/2}}{x^{1/2}} \)

1. **Identify the exponents**: Here, \( a = \frac{1}{2} \) and \( b = \frac{1}{2} \).
2. **Apply the exponent rule**:
\[ y = x^{\frac{1}{2} - \frac{1}{2}} \]
3. **Subtract the exponents**:
\[ y = x^{0} \]
4. **Conclusion**: Therefore, \( n = 0 \).

#### (iv) \( y = \frac{x^{3/2}}{\sqrt{x}} \)

1. **Identify the exponents**: Here, \( a = \frac{3}{2} \) and \( b = \frac{1}{2} \) (since \( \sqrt{x} = x^{1/2} \)).
2. **Apply the exponent rule**:
\[ y = x^{\frac{3}{2} - \frac{1}{2}} \]
3. **Subtract the exponents**:
\[ y = x^{\frac{3}{2} - \frac{1}{2}} = x^{\frac{2}{2}} = x^{1} \]
4. **Conclusion**: Therefore, \( n = 1 \).

### Summary of Results:
- (i) \( n = \frac{4}{3} \)
- (ii) \( n = \frac{1}{2} \)
- (iii) \( n = 0 \)
- (iv) \( n = 1 \)