Which of the following is equivalent to `x^((5)/(7))`, for all values of x?

Options

A. `(5)/(x^(7))`
B. `(1)/(x^(2))`
C. `sqrt(5)(x^(7))`
D. `sqrt(7)(x^(5))`

Explanation

To find an equivalent expression for \( x^{\frac{5}{7}} \), we can use the properties of fractional exponents. Here’s the step-by-step solution:

### Step 1: Understand the Fractional Exponent
The expression \( x^{\frac{5}{7}} \) can be interpreted using the rules of exponents. The numerator (5) represents the power, and the denominator (7) represents the root.

### Step 2: Rewrite the Fractional Exponent
Using the property of exponents, we can rewrite \( x^{\frac{5}{7}} \) as:
\[ x^{\frac{5}{7}} = \left( x^5 \right)^{\frac{1}{7}} \]
This means we take \( x \) raised to the 5th power and then take the 7th root of that result.

### Step 3: Express in Radical Form
Using the radical notation, we can express the above equation as:
\[ x^{\frac{5}{7}} = \sqrt[7]{x^5} \]
This indicates that we are taking the 7th root of \( x^5 \).

### Step 4: Conclusion
Thus, the equivalent expression for \( x^{\frac{5}{7}} \) is:
\[ \sqrt[7]{x^5} \]

### Final Answer
The correct option that represents \( x^{\frac{5}{7}} \) for all values of \( x \) is \( \sqrt[7]{x^5} \).

---