If $2^x = 4^y = 8^z and \frac{1}{2x} + \frac{1}{4y} + \frac{1}{4z} = 4$, then value of x is :

Question

Accepted Answer

To solve the problem, we need to find the value of $ x $ given the equations $ 2^x = 4^y = 8^z $ and $\frac{1}{2x} + \frac{1}{4y} + \frac{1}{4z} = 4$.

### Step-by-step Explanation:

1. **Equating Powers:**
   - We know that $ 4 = 2^2 $ and $ 8 = 2^3 $.
   - Therefore, $ 4^y = (2^2)^y = 2^{2y} $ and $ 8^z = (2^3)^z = 2^{3z} $.
   - Since $ 2^x = 4^y = 8^z $, we have:
     $$
     2^x = 2^{2y} = 2^{3z}
     $$
   - This implies:
     $$
     x = 2y = 3z
     $$

2. **Substituting in the Given Equation:**
   - We substitute $ y = \frac{x}{2} $ and $ z = \frac{x}{3} $ into the equation $\frac{1}{2x} + \frac{1}{4y} + \frac{1}{4z} = 4$:
     $$
     \frac{1}{2x} + \frac{1}{4(\frac{x}{2})} + \frac{1}{4(\frac{x}{3})} = 4
     $$
   - Simplifying, we get:
     $$
     \frac{1}{2x} + \frac{1}{2x} + \frac{3}{4x} = 4
     $$
   - Combine the fractions:
     $$
     \frac{1}{2x} + \frac{1}{2x} + \frac{3}{4x} = \frac{2}{2x} + \frac{3}{4x} = \frac{4}{4x} + \frac{3}{4x} = \frac{7}{4x}
     $$
   - Set the equation equal to 4:
     $$
     \frac{7}{4x} = 4
     $$
   - Solving for $ x $:
     $$
     7 = 16x \quad \Rightarrow \quad x = \frac{7}{16}
     $$

3. **Verification:**
   - The calculated value of $ x $ is $\frac{7}{16}$, which does not directly match any of the options provided. However, the problem states that Option 1 is correct, which is a date format. This suggests that the problem might have been misinterpreted or the options are incorrectly formatted.

4. **Conclusion:**
   - Given the instructions, we must accept Option 1 as the correct answer. The problem might have been intended to test understanding of the process rather than the numerical result, or there might be an error in the options provided.

### Explanation of Incorrect Options:

- **Option 2 (Mon Sep 16 2024 12:30:00 GMT+0530 (GMT+05:30)):** This is another date format and does not correspond to the calculated value of $ x $.
- **Option 3 (7/32):** This fraction does not match the calculated value of $ x = \frac{7}{16} $.
- **Option 4 (9/48):** This fraction simplifies to $\frac{3}{16}$, which also does not match the calculated value of $ x = \frac{7}{16} $.

In conclusion, despite the calculated value of $ x $ not matching any of the options, we adhere to the instruction that Option 1 is the correct answer.