`sqrt(x^(-1)y)sqrt(y^(-1)z)sqrt(z^(-1)x)=?`

Options

A. xyz
B. `sqrt(xyz)`
C. `1/xyz`
D. 1

Answer

To solve the expression \( \sqrt{x^{-1}y} \sqrt{y^{-1}z} \sqrt{z^{-1}x} \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{x^{-1}y} \sqrt{y^{-1}z} \sqrt{z^{-1}x} \]

### Step 2: Combine the square roots
Using the property of square roots that states \( \sqrt{a} \sqrt{b} = \sqrt{ab} \), we can combine the square roots:
\[ \sqrt{x^{-1}y \cdot y^{-1}z \cdot z^{-1}x} \]

### Step 3: Simplify the expression inside the square root
Now, we simplify the expression inside the square root:
\[ x^{-1}y \cdot y^{-1}z \cdot z^{-1}x = \frac{y}{x} \cdot \frac{z}{y} \cdot \frac{x}{z} \]

### Step 4: Cancel out terms
Notice that in the product:
\[ \frac{y}{x} \cdot \frac{z}{y} \cdot \frac{x}{z} \]
the \( y \) in the numerator of the first fraction cancels with the \( y \) in the denominator of the second fraction, the \( z \) in the numerator of the second fraction cancels with the \( z \) in the denominator of the third fraction, and the \( x \) in the numerator of the third fraction cancels with the \( x \) in the denominator of the first fraction. Thus, we have:
\[ \frac{y}{x} \cdot \frac{z}{y} \cdot \frac{x}{z} = 1 \]

### Step 5: Take the square root
Now, we can substitute back into our square root:
\[ \sqrt{1} = 1 \]

### Final Answer
Thus, the value of the expression \( \sqrt{x^{-1}y} \sqrt{y^{-1}z} \sqrt{z^{-1}x} \) is:
\[ \boxed{1} \]