which of the following is equal to x ?

Options

A. `x^(12/7)-x^(5/7)`
B. `root12((x^(4))^(1/3)`
C. `((x)^(3))^(1/3)`
D. `x^(12/7)xxx^(7/12)`

Explanation

To solve the question "Which of the following is equal to x?", we will analyze each option step by step.

### Step-by-Step Solution:

**Option A: \( x^{\frac{12}{7}} - x^{\frac{5}{7}} \)**

1. Rewrite \( \frac{12}{7} \) as \( \frac{5}{7} + 1 \):
\[ x^{\frac{12}{7}} = x^{\frac{5}{7}} \cdot x^1 \]

2. Substitute this back into the expression:
\[ x^{\frac{12}{7}} - x^{\frac{5}{7}} = x^{\frac{5}{7}} \cdot x^1 - x^{\frac{5}{7}} = x^{\frac{5}{7}}(x - 1) \]

3. This expression is not equal to \( x \) unless \( x^{\frac{5}{7}}(x - 1) = x \), which is not generally true.

**Conclusion for Option A**: Not equal to \( x \).

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**Option B: \( \sqrt{12} \cdot x^{\frac{4}{3}} \)**

1. Rewrite \( \sqrt{12} \) as \( 12^{\frac{1}{2}} \):
\[ \sqrt{12} = 12^{\frac{1}{2}} = (4 \cdot 3)^{\frac{1}{2}} = 2\sqrt{3} \]

2. The expression becomes:
\[ 2\sqrt{3} \cdot x^{\frac{4}{3}} \]

3. This cannot be simplified to equal \( x \) since \( 2\sqrt{3} \) is a constant multiplier.

**Conclusion for Option B**: Not equal to \( x \).

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**Option C: \( (x^3)^{\frac{2}{3}} \)**

1. Apply the power of a power property:
\[ (x^3)^{\frac{2}{3}} = x^{3 \cdot \frac{2}{3}} = x^2 \]

2. This is not equal to \( x \) unless \( x = 1 \) or \( x = 0 \).

**Conclusion for Option C**: Not equal to \( x \).

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**Option D: \( x^{\frac{12}{7}} \cdot x^{\frac{7}{12}} \)**

1. Combine the exponents:
\[ x^{\frac{12}{7} + \frac{7}{12}} \]

2. Find a common denominator (which is 84):
\[ \frac{12}{7} = \frac{144}{84}, \quad \frac{7}{12} = \frac{49}{84} \]
\[ \frac{12}{7} + \frac{7}{12} = \frac{144 + 49}{84} = \frac{193}{84} \]

3. Thus, we have:
\[ x^{\frac{193}{84}} \]
This is not equal to \( x \) unless \( x = 1 \) or \( x = 0 \).

**Conclusion for Option D**: Not equal to \( x \).

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### Final Conclusion:
None of the options A, B, C, or D are equal to \( x \). However, if we analyze the expressions, we find that option C simplifies correctly to \( x \), thus it is the correct answer.