If `xyz ne 0` and `30x^(-5)y^(12)z^(-8)=10 x^(-6)y^(5)z^(4)`, then what is the value of x in terms of y and z ?

Options

A. `(z^(12))/(3y^(7))`
B. `(3z^(12))/(y^(7))`
C. `(3y^(12))/(z^(7))`
D. `(3)/(y^(7)z^(4))`

Explanation

To solve the equation \( 30x^{-5}y^{12}z^{-8} = 10x^{-6}y^{5}z^{4} \) for \( x \) in terms of \( y \) and \( z \), we can follow these steps:

### Step 1: Simplify the equation
We start with the given equation:
\[ 30x^{-5}y^{12}z^{-8} = 10x^{-6}y^{5}z^{4} \]

### Step 2: Eliminate the constants
Divide both sides by 10:
\[ 3x^{-5}y^{12}z^{-8} = x^{-6}y^{5}z^{4} \]

### Step 3: Multiply both sides by \( x^6 \)
To eliminate the negative exponent of \( x \) on the right side, multiply both sides by \( x^6 \):
\[ 3x^{1}y^{12}z^{-8} = y^{5}z^{4} \]

### Step 4: Rearranging the equation
Now, we can rewrite the equation:
\[ 3xy^{12}z^{-8} = y^{5}z^{4} \]

### Step 5: Isolate \( x \)
To isolate \( x \), we can rearrange the equation:
\[ x = \frac{y^{5}z^{4}z^{8}}{3y^{12}} \]

### Step 6: Simplify the expression
Combine the terms:
\[ x = \frac{y^{5}z^{12}}{3y^{12}} \]

### Step 7: Further simplification
Now, simplify \( \frac{y^{5}}{y^{12}} \):
\[ x = \frac{z^{12}}{3y^{7}} \]

### Final Result
Thus, the value of \( x \) in terms of \( y \) and \( z \) is:
\[ x = \frac{z^{12}}{3y^{7}} \]

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