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

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

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. **Expressing Powers in Terms of Base 2:** - 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} $. 2. **Equating the Powers:** - Since $ 2^x = 4^y = 8^z $, we have: \[ 2^x = 2^{2y} = 2^{3z} \] - This implies: \[ x = 2y = 3z \] 3. **Substituting in the Given Equation:** - We have the equation $\frac{1}{2x} + \frac{1}{4y} + \frac{1}{4z} = 4$. - Substitute $ y = \frac{x}{2} $ and $ z = \frac{x}{3} $ into the equation: \[ \frac{1}{2x} + \frac{1}{4(\frac{x}{2})} + \frac{1}{4(\frac{x}{3})} = 4 \] - Simplify the terms: \[ \frac{1}{2x} + \frac{1}{2x} + \frac{3}{4x} = 4 \] - Combine the fractions: \[ \frac{1}{2x} + \frac{1}{2x} + \frac{3}{4x} = \frac{2}{2x} + \frac{3}{4x} = \frac{4 + 3}{4x} = \frac{7}{4x} \] - Set the equation equal to 4: \[ \frac{7}{4x} = 4 \] 4. **Solving for $ x $:** - Multiply both sides by $ 4x $: \[ 7 = 16x \] - Solve for $ x $: \[ x = \frac{7}{16} \] ### Conclusion: - The value of $ x $ is $\frac{7}{16}$, which corresponds to **Option 1**. ### Explanation of Incorrect Options: - **Option 2 ($\frac{7}{17}$)**: This value does not satisfy the equation $\frac{7}{4x} = 4$ when substituted back, as it does not equate to 4. - **Option 3 ($\frac{7}{19}$)**: Similarly, substituting this value into the equation does not satisfy the condition $\frac{7}{4x} = 4$. - **Option 4 ($\frac{7}{23}$)**: This value also fails to satisfy the equation, as it does not result in the required equality. Thus, the only correct solution that satisfies all given conditions is **Option 1: $\frac{7}{16}$**.

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. **Expressing Powers in Terms of Base 2:**
   - 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} $.

2. **Equating the Powers:**
   - Since $ 2^x = 4^y = 8^z $, we have:
     $$
     2^x = 2^{2y} = 2^{3z}
     $$
   - This implies:
     $$
     x = 2y = 3z
     $$

3. **Substituting in the Given Equation:**
   - We have the equation $\frac{1}{2x} + \frac{1}{4y} + \frac{1}{4z} = 4$.
   - Substitute $ y = \frac{x}{2} $ and $ z = \frac{x}{3} $ into the equation:
     $$
     \frac{1}{2x} + \frac{1}{4(\frac{x}{2})} + \frac{1}{4(\frac{x}{3})} = 4
     $$
   - Simplify the terms:
     $$
     \frac{1}{2x} + \frac{1}{2x} + \frac{3}{4x} = 4
     $$
   - Combine the fractions:
     $$
     \frac{1}{2x} + \frac{1}{2x} + \frac{3}{4x} = \frac{2}{2x} + \frac{3}{4x} = \frac{4 + 3}{4x} = \frac{7}{4x}
     $$
   - Set the equation equal to 4:
     $$
     \frac{7}{4x} = 4
     $$

4. **Solving for $ x $:**
   - Multiply both sides by $ 4x $:
     $$
     7 = 16x
     $$
   - Solve for $ x $:
     $$
     x = \frac{7}{16}
     $$

### Conclusion:

- The value of $ x $ is $\frac{7}{16}$, which corresponds to **Option 1**.

### Explanation of Incorrect Options:

- **Option 2 ($\frac{7}{17}$)**: This value does not satisfy the equation $\frac{7}{4x} = 4$ when substituted back, as it does not equate to 4.
- **Option 3 ($\frac{7}{19}$)**: Similarly, substituting this value into the equation does not satisfy the condition $\frac{7}{4x} = 4$.
- **Option 4 ($\frac{7}{23}$)**: This value also fails to satisfy the equation, as it does not result in the required equality.

Thus, the only correct solution that satisfies all given conditions is **Option 1: $\frac{7}{16}$**.