If u : v = v : w, then what is the value of `u^4 : v^4`?

Options

A. `u^2 : Wv`
B. `u^2 : w^2`
C. `u^2 :w^2`
D. `u:w`

Answer

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

1. **Given**: \( u : v = v : w \)

2. **Expressing in Fraction Form**:
From the ratio, we can write:
\[ \frac{u}{v} = \frac{v}{w} \]

3. **Cross Multiplying**:
By cross-multiplying, we get:
\[ u \cdot w = v \cdot v \]
or
\[ u \cdot w = v^2 \]

4. **Finding \( u^4 : v^4 \)**:
We need to find the value of \( u^4 : v^4 \). We can express this as:
\[ \frac{u^4}{v^4} \]

5. **Substituting \( v^2 \)**:
From \( v^2 = u \cdot w \), we can express \( v^4 \) as:
\[ v^4 = (v^2)^2 = (u \cdot w)^2 = u^2 \cdot w^2 \]

6. **Substituting Back**:
Now substituting \( v^4 \) back into our ratio:
\[ \frac{u^4}{v^4} = \frac{u^4}{u^2 \cdot w^2} \]

7. **Simplifying the Ratio**:
Simplifying this gives:
\[ \frac{u^4}{u^2 \cdot w^2} = \frac{u^4}{u^2} \cdot \frac{1}{w^2} = \frac{u^{4-2}}{w^2} = \frac{u^2}{w^2} \]

8. **Final Ratio**:
Therefore, the final ratio \( u^4 : v^4 \) is:
\[ u^2 : w^2 \]

### Final Answer:
The value of \( u^4 : v^4 \) is \( u^2 : w^2 \).