`((a)/(b))^(x+y+z)div[(sqrt(a/b))^(-x)xx(sqrta/b)^(-y)xx(sqrt(a/b))^(-z)]=`________.

Options

A. `[a^3//b^3]^(x+y+z) `
B. `[a^2//b^2]^(x+y+z)`
C. `[a//b]^(x+y+z)//2`
D. `[a//b]^3(x+y+z)//2`

Explanation

To solve the expression \(\frac{\left(\frac{a}{b}\right)^{x+y+z}}{\left(\sqrt{\frac{a}{b}}\right)^{-x} \cdot \left(\sqrt{\frac{a}{b}}\right)^{-y} \cdot \left(\sqrt{\frac{a}{b}}\right)^{-z}}\), we will follow these steps:

### Step 1: Rewrite the square roots as powers
The square root can be expressed as a power of \(\frac{1}{2}\):
\[ \sqrt{\frac{a}{b}} = \left(\frac{a}{b}\right)^{\frac{1}{2}} \]
Thus, we can rewrite the denominator:
\[ \left(\sqrt{\frac{a}{b}}\right)^{-x} = \left(\frac{a}{b}\right)^{-\frac{x}{2}}, \quad \left(\sqrt{\frac{a}{b}}\right)^{-y} = \left(\frac{a}{b}\right)^{-\frac{y}{2}}, \quad \left(\sqrt{\frac{a}{b}}\right)^{-z} = \left(\frac{a}{b}\right)^{-\frac{z}{2}} \]

### Step 2: Combine the terms in the denominator
Now, we can combine the terms in the denominator:
\[ \left(\sqrt{\frac{a}{b}}\right)^{-x} \cdot \left(\sqrt{\frac{a}{b}}\right)^{-y} \cdot \left(\sqrt{\frac{a}{b}}\right)^{-z} = \left(\frac{a}{b}\right)^{-\frac{x}{2}} \cdot \left(\frac{a}{b}\right)^{-\frac{y}{2}} \cdot \left(\frac{a}{b}\right)^{-\frac{z}{2}} = \left(\frac{a}{b}\right)^{-\left(\frac{x+y+z}{2}\right)} \]

### Step 3: Rewrite the entire expression
Now we can rewrite the entire expression:
\[ \frac{\left(\frac{a}{b}\right)^{x+y+z}}{\left(\frac{a}{b}\right)^{-\left(\frac{x+y+z}{2}\right)}} \]

### Step 4: Apply the property of indices
Using the property of indices \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify:
\[ \left(\frac{a}{b}\right)^{(x+y+z) - \left(-\frac{x+y+z}{2}\right)} = \left(\frac{a}{b}\right)^{(x+y+z) + \frac{x+y+z}{2}} \]

### Step 5: Combine the powers
Now, we can combine the powers:
\[ (x+y+z) + \frac{x+y+z}{2} = \frac{2(x+y+z)}{2} + \frac{x+y+z}{2} = \frac{3(x+y+z)}{2} \]

### Final Result
Thus, the expression simplifies to:
\[ \left(\frac{a}{b}\right)^{\frac{3(x+y+z)}{2}} \]