Simplify : `((xyz)^4)/((x^(-2) y^(3) )^(-3) ((1)/(z^2) )^6) ( x ne 0, y ne 0 , z ne 0)`.

Answer

To simplify the expression \(\frac{(xyz)^4}{(x^{-2} y^{3})^{-3} \left(\frac{1}{z^2}\right)^{6}}\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{(xyz)^4}{(x^{-2} y^{3})^{-3} \left(\frac{1}{z^2}\right)^{6}} \]

### Step 2: Simplify the numerator
The numerator \((xyz)^4\) can be expanded using the property of exponents:
\[ (xyz)^4 = x^4 y^4 z^4 \]

### Step 3: Simplify the denominator
Now, simplify the denominator:
1. For \((x^{-2} y^{3})^{-3}\):
\[ (x^{-2} y^{3})^{-3} = x^{6} y^{-9} \]
(using the property \(a^{-m} = \frac{1}{a^m}\))

2. For \(\left(\frac{1}{z^2}\right)^{6}\):
\[ \left(\frac{1}{z^2}\right)^{6} = z^{-12} \]

Combining these, the denominator becomes:
\[ x^{6} y^{-9} z^{-12} \]

### Step 4: Combine the numerator and denominator
Now we can write the entire expression:
\[ \frac{x^4 y^4 z^4}{x^{6} y^{-9} z^{-12}} \]

### Step 5: Apply the property of exponents
Using the property \(\frac{a^m}{a^n} = a^{m-n}\), we simplify each variable:
1. For \(x\):
\[ x^{4 - 6} = x^{-2} \]

2. For \(y\):
\[ y^{4 - (-9)} = y^{4 + 9} = y^{13} \]

3. For \(z\):
\[ z^{4 - (-12)} = z^{4 + 12} = z^{16} \]

### Step 6: Write the final expression
Combining all these results, we have:
\[ \frac{x^{-2} y^{13} z^{16}}{1} = \frac{y^{13} z^{16}}{x^{2}} \]

Thus, the simplified expression is:
\[ \frac{y^{13} z^{16}}{x^{2}} \]