Which is equivalent to `x^(-2/3)` =4 ?

Options

A. `x^(2/3)=1/4`
B. `x^(2/3)=-4`
C. `x^(-2/3)=-1/4`
D. `x^(3/2)=1/4`

Answer

To solve the equation \( x^{-\frac{2}{3}} = 4 \), we can follow these steps:

### Step 1: Rewrite the equation
Start with the original equation:
\[ x^{-\frac{2}{3}} = 4 \]

### Step 2: Take the reciprocal
Since the exponent is negative, we can rewrite the equation using the reciprocal:
\[ \frac{1}{x^{\frac{2}{3}}} = 4 \]

### Step 3: Cross-multiply
To eliminate the fraction, we can cross-multiply:
\[ 1 = 4 \cdot x^{\frac{2}{3}} \]

### Step 4: Isolate \( x^{\frac{2}{3}} \)
Now, divide both sides by 4 to isolate \( x^{\frac{2}{3}} \):
\[ x^{\frac{2}{3}} = \frac{1}{4} \]

### Step 5: Raise both sides to the power of \( \frac{3}{2} \)
To solve for \( x \), raise both sides to the power of \( \frac{3}{2} \):
\[ x = \left(\frac{1}{4}\right)^{\frac{3}{2}} \]

### Step 6: Simplify the right side
Calculating \( \left(\frac{1}{4}\right)^{\frac{3}{2}} \):
\[ \left(\frac{1}{4}\right)^{\frac{3}{2}} = \frac{1^3}{4^{\frac{3}{2}}} = \frac{1}{(2^2)^{\frac{3}{2}}} = \frac{1}{2^{3}} = \frac{1}{8} \]

### Final Answer
Thus, the value of \( x \) is:
\[ x = \frac{1}{8} \]