how to do 2x to the power of 2/3 to the power of -3

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!

Learn how to simplify the expression (2x^(2/3))^(-3) step-by-step using exponent rules. This problem covers concepts of exponents, power rules, and negative exponents, making it suitable for students in Grades 8-10.

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