21. \( 2 x^{\frac{1}{2}}-x^{\frac{1}{4}}=0 \)

To solve the equation \( 2 x^{\frac{1}{2}} - x^{\frac{1}{4}} = 0 \), first, you can factor out \( x^{\frac{1}{4}} \): \[ x^{\frac{1}{4}}(2x^{\frac{1}{4}} - 1) = 0. \] This gives us two possible solutions: \( x^{\frac{1}{4}} = 0 \), which means \( x = 0 \), and the second equation \( 2x^{\frac{1}{4}} - 1 = 0 \), leading to \( x^{\frac{1}{4}} = \frac{1}{2} \). Elevating both sides to the power of 4 gives \( x = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \). Thus, the solutions are \( x = 0 \) and \( x = \frac{1}{16} \). For those diving deeper into similar equations, it's crucial to familiarize yourself with fractional exponents and their manipulations. They can often be transformed into polynomial equations, which are generally easier to handle. Remember, practice is key—try out other similar forms, adapting the factoring method! In practice, these types of algebraic equations appear frequently in physics and engineering, especially when dealing with equations that model phenomena like projectile motion or electrical circuits. Solving such equations accurately can lead to insightful conclusions about the behavior of these systems. Keep your math toolkit sharp – it'll come in handy the next time you're graphing the motion of a launched object or designing a circuit!