The expression `(x^(2))/(sqrt(x^(2)))` is equivalent to

Options

A. `root(3)(y)`
B. `(1)/(sqrt(x))`
C. `sqrt(x^2)`
D. `(1)/(root(3)(x^(2)))`

Answer

To simplify the expression \(\frac{x^2}{\sqrt{x^2}}\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
The expression is \(\frac{x^2}{\sqrt{x^2}}\). Here, \(x^2\) is the numerator and \(\sqrt{x^2}\) is the denominator.

2. **Simplify the Denominator**:
We know that \(\sqrt{x^2} = |x|\). This is because the square root of a square gives the absolute value of the base. Therefore, we can rewrite the expression as:
\[ \frac{x^2}{\sqrt{x^2}} = \frac{x^2}{|x|} \]

3. **Simplify the Fraction**:
Now we simplify the fraction \(\frac{x^2}{|x|}\). Since \(x^2 = x \cdot x\), we can express this as:
\[ \frac{x \cdot x}{|x|} \]
We can further simplify this by recognizing that \(|x|\) can be either \(x\) (if \(x \geq 0\)) or \(-x\) (if \(x < 0\)).

4. **Consider Cases**:
- **Case 1**: If \(x \geq 0\), then \(|x| = x\):
\[ \frac{x^2}{|x|} = \frac{x^2}{x} = x \]
- **Case 2**: If \(x < 0\), then \(|x| = -x\):
\[ \frac{x^2}{|x|} = \frac{x^2}{-x} = -x \]

5. **Final Result**:
Therefore, the expression \(\frac{x^2}{\sqrt{x^2}}\) simplifies to:
\[ x \quad \text{(if \(x \geq 0\))} \quad \text{or} \quad -x \quad \text{(if \(x < 0\))} \]
In summary, we can express the simplified form as:
\[ \frac{x^2}{\sqrt{x^2}} = |x| \]

### Final Answer:
The expression \(\frac{x^2}{\sqrt{x^2}}\) is equivalent to \(|x|\).