Let's simplify the given expression step by step to find the exponent of $x$ when written as a single power.

Expression:

$\left( \frac{x^{-1/3} x^{1/6}}{x^{1/4} x^{-1/2}} \right)^{-1/3}$

Step 1: Simplify the numerator and denominator

• Numerator: $x^{-1/3} \cdot x^{1/6} = x^{-1/3 + 1/6} = x^{-2/6 + 1/6} = x^{-1/6}$

• Denominator: $x^{1/4} \cdot x^{-1/2} = x^{1/4 - 1/2} = x^{1/4 - 2/4} = x^{-1/4}$

Step 2: Combine the numerator and denominator

$\frac{x^{-1/6}}{x^{-1/4}} = x^{-1/6 - (-1/4)} = x^{-1/6 + 1/4}$

Now, we find the common denominator: $\frac{-1}{6} + \frac{1}{4} = \frac{-2}{12} + \frac{3}{12} = \frac{1}{12}$

So the simplified expression becomes: $x^{1/12}$

Step 3: Apply the outer exponent

Now, raise this to the power of $-1/3$: $\left( x^{1/12} \right)^{-1/3} = x^{1/12 \cdot (-1/3)} = x^{-1/36}$

Final Answer:

The exponent of $x$ is $-\frac{1}{36}$.

Would you like further details on any of these steps? Here are five related questions you can consider:

1. How do you combine exponents when multiplying powers of the same base?
2. How does raising a power to another power affect the exponents?
3. How do you find the common denominator when adding fractions?
4. Why does dividing powers involve subtracting exponents?
5. What are some common mistakes when simplifying expressions with fractional exponents?

Tip: When dealing with exponents, always remember that multiplying powers with the same base means adding exponents, and dividing powers means subtracting exponents.