If $2^x = 4^y = 8^z$ and $$ \left(\frac{1}{2x} + \frac{1}{4y} + \frac{1}{6z}\right) = \frac{24}{7} $$ then the value of $z$ is

### Concept & Formula When given a chain of exponential equalities with bases that are powers of the same number (e.g., $2, 4, 8$), convert all bases to the smallest common prime base. Then, express all variables in terms of the specific variable you need to find. ### Step-by-Step Solution * **Step 1:** Express the exponential chain using a common base of $2$. $ 2^x = (2^2)^y = (2^3)^z $ $ 2^x = 2^{2y} = 2^{3z} $ * **Step 2:** Equate the exponents to find the relationship between $x, y,$ and $z$. $ x = 2y = 3z $ * **Step 3:** The problem asks for the value of $z$. Therefore, convert all terms in the given fractional equation into terms of $z$. We know: $ x = 3z $ $\rightarrow$ $ 2x = 6z $ $ 2y = 3z $ $\rightarrow$ Multiply by 2 $\rightarrow$ $ 4y = 6z $ * **Step 4:** Substitute these $z$-based values into the given equation. $$ \frac{1}{2x} + \frac{1}{4y} + \frac{1}{6z} = \frac{24}{7} $$ $$ \frac{1}{6z} + \frac{1}{6z} + \frac{1}{6z} = \frac{24}{7} $$ * **Step 5:** Simplify and solve for $z$. $$ \frac{3}{6z} = \frac{24}{7} $$ $$ \frac{1}{2z} = \frac{24}{7} $$ Cross-multiply: $ 7 = 48z $ $ z = \frac{7}{48} $ ### Exam Strategy & Shortcut Notice the symmetry in the fractional equation once you find $x = 2y = 3z$. The denominators $2x, 4y,$ and $6z$ look different, but checking their base relationships ($2(3z) = 6z$, $2(2(2y)) \dots$ wait, $2y = 3z \rightarrow 4y = 6z$) reveals they are completely identical. Recognizing instantly that the equation is just $3 \times (\text{one term}) = \frac{24}{7}$ allows you to jump straight to $\frac{1}{2z} = \frac{24}{7}$, solving the problem mentally in under 30 seconds. ### Common Pitfall A frequent mistake is solving for the wrong variable (like finding $x$ instead of $z$) and then scrambling to convert it at the end, which introduces fraction arithmetic errors. Always substitute so that the only remaining variable is the one requested by the question. ### Final Answer Therefore, the correct answer is **$\frac{7}{48}$**.