We are given the equation \( |x^{-\frac{2}{3}}| = 2 \). Let's solve this step-by-step:

### Step 1: Remove the absolute value
Since \( |a| = b \) implies \( a = b \) or \( a = -b \), we remove the absolute value as follows:

\[
x^{-\frac{2}{3}} = 2 \quad \text{or} \quad x^{-\frac{2}{3}} = -2
\]

### Step 2: Solve each case separately

#### Case 1: \( x^{-\frac{2}{3}} = 2 \)

1. Rewrite the equation as a reciprocal:
   \[
   \frac{1}{x^{\frac{2}{3}}} = 2
   \]

2. Take the reciprocal of both sides:
   \[
   x^{\frac{2}{3}} = \frac{1}{2}
   \]

3. Raise both sides to the power of \( \frac{3}{2} \) to cancel the exponent:
   \[
   x = \left( \frac{1}{2} \right)^{\frac{3}{2}} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}
   \]

#### Case 2: \( x^{-\frac{2}{3}} = -2 \)

This case has no real solution because \( x^{-\frac{2}{3}} \), which is a reciprocal power, cannot equal a negative number (since raising a real number to a fractional power results in a positive number).

### Final Solution
The only real solution is:
\[
x = \frac{\sqrt{2}}{4}
\]

Would you like more details or have any questions?

Here are 5 related questions you can explore:
1. How do we handle equations involving fractional exponents?
2. What happens when the base of an exponent is negative?
3. Can we solve \( |x^{a}| = b \) when \( b \) is negative?
4. What are the properties of absolute values and how are they applied in equations?
5. What is the meaning of reciprocal powers in algebra?

**Tip**: When solving equations with absolute values, always check for extraneous solutions after removing the absolute value!

Learn how to solve the equation |x^-2/3| = 2 with a detailed step-by-step solution. This problem explores algebraic concepts including absolute values, fractional exponents, and reciprocal powers. Ideal for students in Grades 10-12.

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