If `x^(-(1)/(2))=(1)/(8)`, what is the value of `x^((2)/(3))`?

Answer

To solve the equation \( x^{-\frac{1}{2}} = \frac{1}{8} \) and find the value of \( x^{\frac{2}{3}} \), we can follow these steps:

### Step 1: Rewrite the equation
The equation \( x^{-\frac{1}{2}} = \frac{1}{8} \) can be rewritten using the property of exponents:
\[ \frac{1}{\sqrt{x}} = \frac{1}{8} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ \sqrt{x} = 8 \]

### Step 3: Square both sides
To eliminate the square root, we square both sides:
\[ x = 8^2 \]
Calculating \( 8^2 \):
\[ x = 64 \]

### Step 4: Find \( x^{\frac{2}{3}} \)
Now we need to find \( x^{\frac{2}{3}} \):
\[ x^{\frac{2}{3}} = (64)^{\frac{2}{3}} \]

### Step 5: Simplify \( (64)^{\frac{2}{3}} \)
We can break this down into two parts:
1. First, find \( 64^{\frac{1}{3}} \):
\[ 64^{\frac{1}{3}} = 4 \quad \text{(since \( 4^3 = 64 \))} \]
2. Now square the result:
\[ (64^{\frac{1}{3}})^2 = 4^2 = 16 \]

### Final Answer
Thus, the value of \( x^{\frac{2}{3}} \) is:
\[ \boxed{16} \]

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