If `xgt1` and `(sqrt(x^(3)))/(x^(2))=x^(n)`, what is the value of n?

Options

A. `-(3)/(2)`
B. `-1`
C. `-(1)/(2)`
D. `(1)/(2)`

Answer

To solve the equation \(\frac{\sqrt{x^3}}{x^2} = x^n\) for \(n\), we can follow these steps:

### Step 1: Rewrite the square root
The square root of \(x^3\) can be expressed using exponent notation:
\[ \sqrt{x^3} = x^{3/2} \]

### Step 2: Substitute into the equation
Now, substitute this back into the equation:
\[ \frac{x^{3/2}}{x^2} = x^n \]

### Step 3: Simplify the left side
When dividing powers with the same base, we subtract the exponents:
\[ \frac{x^{3/2}}{x^2} = x^{3/2 - 2} \]

### Step 4: Calculate the exponent
Now, calculate \(3/2 - 2\):
\[ 3/2 - 2 = 3/2 - 4/2 = -1/2 \]
So, we have:
\[ x^{-1/2} = x^n \]

### Step 5: Compare exponents
Since the bases are the same, we can set the exponents equal to each other:
\[ n = -\frac{1}{2} \]

### Final Answer
Thus, the value of \(n\) is:
\[ \boxed{-\frac{1}{2}} \]