Value of `{ (root (n) x ^(2)) ^((n)/(2))}^(2)` is

Options

A. `x`
B. `x ^(n//2)`
C. `x ^(2)`
D. `(1)/(x ^(2))`

Answer

To solve the expression \( \left( \sqrt[n]{x^2} \right)^{\frac{n}{2}} \) step by step, we can follow these steps:

### Step 1: Rewrite the expression in index form
The expression \( \sqrt[n]{x^2} \) can be rewritten using exponent notation. The nth root of \( x^2 \) is equivalent to \( x^2 \) raised to the power of \( \frac{1}{n} \):
\[ \sqrt[n]{x^2} = x^{\frac{2}{n}} \]
Thus, we can rewrite the entire expression:
\[ \left( \sqrt[n]{x^2} \right)^{\frac{n}{2}} = \left( x^{\frac{2}{n}} \right)^{\frac{n}{2}} \]

### Step 2: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify the expression:
\[ \left( x^{\frac{2}{n}} \right)^{\frac{n}{2}} = x^{\frac{2}{n} \cdot \frac{n}{2}} \]

### Step 3: Simplify the exponent
Now, we simplify the exponent:
\[ \frac{2}{n} \cdot \frac{n}{2} = \frac{2n}{2n} = 1 \]
Thus, we have:
\[ x^{1} = x \]

### Step 4: Final result
Therefore, the value of the original expression \( \left( \sqrt[n]{x^2} \right)^{\frac{n}{2}} \) simplifies to:
\[ x \]

### Conclusion
The final value of the expression is \( x \).