Solve the equation. \[ \left|\frac{1}{4} z\right|=|4 z| \]

To solve the equation: \[ \left|\frac{1}{4} z\right| = |4 z| \] let's analyze both sides of the equation. 1. **Simplify the Absolute Values:** The absolute value of a complex number (or real number) scaled by a constant can be simplified by taking the absolute value of the constant multiplied by the absolute value of the number. Therefore: \[ \left|\frac{1}{4} z\right| = \frac{1}{4} |z| \] and \[ |4 z| = 4 |z| \] 2. **Set Up the Equation:** Substitute the simplified expressions back into the original equation: \[ \frac{1}{4} |z| = 4 |z| \] 3. **Solve for \(|z|\):** To solve for \(|z|\), subtract \(\frac{1}{4} |z|\) from both sides: \[ \frac{1}{4} |z| - 4 |z| = 0 \implies -\frac{15}{4} |z| = 0 \] The only solution to this equation is: \[ |z| = 0 \] 4. **Interpret the Result:** The absolute value of \(z\) being zero implies that \(z\) itself must be zero. **Conclusion:** The only solution to the equation is \( z = 0 \). **Answer:** \(0\)