If `x^(2)=a^((1)/(3))`, where `xgt0 and agt0`, which of the following gives a in terms of x?

Options

A. `a=(1)/(x^(6))`
B. `a=(2)/(x^(3))`
C. `a=x^((3)/(2)`
D. `a=-x^(6)`

Explanation

To find \( a \) in terms of \( x \) from the equation \( x^2 = a^{-\frac{1}{3}} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^2 = a^{-\frac{1}{3}} \]

### Step 2: Eliminate the negative exponent
To eliminate the negative exponent, we can rewrite \( a^{-\frac{1}{3}} \) as:
\[ x^2 = \frac{1}{a^{\frac{1}{3}}} \]
This means:
\[ x^2 \cdot a^{\frac{1}{3}} = 1 \]

### Step 3: Isolate \( a^{\frac{1}{3}} \)
Now, we can isolate \( a^{\frac{1}{3}} \):
\[ a^{\frac{1}{3}} = \frac{1}{x^2} \]

### Step 4: Cube both sides
To solve for \( a \), we need to eliminate the exponent \( \frac{1}{3} \) by cubing both sides:
\[ a = \left(\frac{1}{x^2}\right)^3 \]

### Step 5: Simplify the expression
Now we simplify the right-hand side:
\[ a = \frac{1}{x^6} \]

Thus, the final expression for \( a \) in terms of \( x \) is:
\[ a = \frac{1}{x^6} \]