If `x^(n)` is the simplified form of the expression above, what is the value of n?

Answer

To solve the problem, we need to simplify the expression given and find the value of \( n \) in \( x^n \).

### Step-by-Step Solution:

1. **Rewrite the Expression**:
The given expression is \( x^{\frac{1}{3}} \cdot x^{\frac{5}{2}} \cdot x^1 \div x^{\frac{1}{2}} \).

2. **Convert Roots to Exponents**:
- The cube root of \( x \) can be expressed as \( x^{\frac{1}{3}} \).
- The square root of \( x \) can be expressed as \( x^{\frac{1}{2}} \).
- Thus, we rewrite the expression as:
\[ x^{\frac{1}{3}} \cdot x^{\frac{5}{2}} \cdot x^1 \cdot x^{-\frac{1}{2}} \]

3. **Combine the Exponents**:
According to the laws of exponents, when multiplying like bases, we add the exponents:
\[ x^{\left(\frac{1}{3} + \frac{5}{2} + 1 - \frac{1}{2}\right)} \]

4. **Finding a Common Denominator**:
To add the fractions, we need a common denominator. The denominators are 3, 2, and 2. The least common denominator is 6.
- Convert each term:
- \( \frac{1}{3} = \frac{2}{6} \)
- \( \frac{5}{2} = \frac{15}{6} \)
- \( 1 = \frac{6}{6} \)
- \( \frac{1}{2} = \frac{3}{6} \)

5. **Add the Exponents**:
Now we can add the converted fractions:
\[ \frac{2}{6} + \frac{15}{6} + \frac{6}{6} - \frac{3}{6} = \frac{2 + 15 + 6 - 3}{6} = \frac{20}{6} \]

6. **Simplify the Result**:
The fraction \( \frac{20}{6} \) can be simplified:
\[ \frac{20}{6} = \frac{10}{3} \]

7. **Final Expression**:
Therefore, the simplified form of the expression is:
\[ x^{\frac{10}{3}} \]

8. **Conclusion**:
Thus, the value of \( n \) is:
\[ n = \frac{10}{3} \]