The expression you've given is $(2x^{\frac{2}{3}})^{-3}$. To simplify this, we can break it down step by step:

Step 1: Apply the power of a power rule

The power of a power rule states that when you raise a power to another power, you multiply the exponents. Here's how it works:

$(2x^{\frac{2}{3}})^{-3} = 2^{-3} \cdot (x^{\frac{2}{3}})^{-3}$

Step 2: Simplify each term

Now simplify each part:

• For $2^{-3}$, calculate the power of 2: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

• For $(x^{\frac{2}{3}})^{-3}$, multiply the exponents $\frac{2}{3} \times (-3)$: $x^{\frac{2}{3} \times -3} = x^{-2}$

Step 3: Combine the results

Now combine the results: $\frac{1}{8} \cdot x^{-2} = \frac{1}{8x^2}$

Final Answer:

The simplified form of $(2x^{\frac{2}{3}})^{-3}$ is: $\frac{1}{8x^2}$

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to practice:

1. Simplify $(3x^{\frac{1}{2}})^{-4}$.
2. What is the value of $(5x^{\frac{3}{4}})^{-2}$?
3. Simplify $(4x^{\frac{5}{6}})^{-3}$.
4. How do you simplify $(2x^{-1})^{-3}$?
5. Simplify $(6x^{\frac{2}{5}})^{\frac{5}{2}}$.

Tip: When raising a term to a negative power, you can flip the term and change the exponent to positive!