Which of the following expressions is equivalent to `-x^(1/4)` ?

Options

A. `-(1)/(4x)`
B. `-(1)/(x ^(4))`
C. `-root4x`
D. `(1)/(root4 (-x))`

Answer

To determine which of the following expressions is equivalent to \(-x^{1/4}\), we can follow these steps:

### Step 1: Understand the expression
The expression \(-x^{1/4}\) indicates that we have a negative sign in front of \(x\) raised to the power of \(\frac{1}{4}\).

### Step 2: Rewrite the fractional exponent
Recall that a fractional exponent can be rewritten in radical form. Specifically, \(x^{1/4}\) can be expressed as the fourth root of \(x\):
\[ x^{1/4} = \sqrt[4]{x} \]

### Step 3: Apply the negative sign
Now, we include the negative sign in front of the expression:
\[ -x^{1/4} = -\sqrt[4]{x} \]

### Step 4: Compare with the options
Now we need to compare \(-\sqrt[4]{x}\) with the given options:
- a) \(-\frac{1}{4x}\)
- b) \(-\frac{1}{x^4}\)
- c) \(-\sqrt[4]{x}\)
- d) \(\frac{1}{\sqrt[4]{-x}}\)

From our simplification, we see that \(-\sqrt[4]{x}\) matches option c.

### Conclusion
Thus, the expression equivalent to \(-x^{1/4}\) is:
\[ \text{Option c: } -\sqrt[4]{x} \]

---