If x:y=y:z then what is the value of `x^(3) : y^(3)?`

Options

A. `x^(2) : yz`
B. `x^(3) : y z`
C. `x:z`
D. `z^(2) : xy`

Answer

To solve the problem, we start with the given ratio:

1. **Given**: \( x : y = y : z \)

2. **Expressing as fractions**: This can be written as:
\[ \frac{x}{y} = \frac{y}{z} \]

3. **Cross-multiplying**: By cross-multiplying, we get:
\[ x \cdot z = y \cdot y \quad \text{or} \quad xz = y^2 \]

4. **Rearranging**: From the equation \( xz = y^2 \), we can express \( y^2 \) in terms of \( x \) and \( z \):
\[ y^2 = xz \]

5. **Multiplying both sides by y**: To find \( y^3 \), we multiply both sides by \( y \):
\[ y^3 = y \cdot y^2 = y \cdot (xz) = xyz \]

6. **Finding the ratio \( \frac{x^3}{y^3} \)**: Now we can express \( \frac{x^3}{y^3} \):
\[ \frac{x^3}{y^3} = \frac{x^3}{xyz} \]

7. **Simplifying the ratio**: This simplifies to:
\[ \frac{x^3}{xyz} = \frac{x^2}{z} \]

8. **Expressing in terms of ratio**: Therefore, we can express this as:
\[ x^3 : y^3 = x^2 : yz \]

Thus, the final answer is:
\[ x^3 : y^3 = x^2 : yz \]

### Answer:
The value of \( x^3 : y^3 \) is \( x^2 : yz \).