To solve the equation:

$x^{\frac{1}{2}} - 4x^{\frac{1}{4}} = 0$

we can approach it by simplifying the terms with substitution. Here’s a step-by-step solution:

Step 1: Substitute a New Variable

Let: $u = x^{\frac{1}{4}}$ Then: $u^2 = x^{\frac{1}{2}}$

This substitution changes the equation to: $u^2 - 4u = 0$

Step 2: Factor the Equation

The equation $u^2 - 4u = 0$ can be factored as: $u(u - 4) = 0$

Step 3: Solve for $u$

Set each factor equal to zero:

1. $u = 0$
2. $u - 4 = 0 \Rightarrow u = 4$

Step 4: Substitute Back for $x$

Now, recall that $u = x^{\frac{1}{4}}$. Substitute back to find $x$ for each solution of $u$.

1. If $u = 0$: $x^{\frac{1}{4}} = 0 \Rightarrow x = 0$

2. If $u = 4$: $x^{\frac{1}{4}} = 4 \Rightarrow x = 4^4 = 256$

Final Answer

The solutions to the equation are: $x = 0 \quad \text{and} \quad x = 256$

Would you like more details on any step?

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Here are five related questions for practice:

1. Solve $x^{\frac{1}{3}} - 3x^{\frac{1}{6}} = 0$.
2. Find all solutions to $x^{\frac{2}{3}} - 5x^{\frac{1}{3}} = 0$.
3. Solve the equation $y^2 - 6y = 0$ using factoring.
4. What are the real solutions to $z^2 - 2z = 0$?
5. Solve $x^{\frac{1}{5}} - 2x^{\frac{1}{10}} = 0$.

Tip: When faced with fractional exponents, substitution can simplify the problem and reveal hidden factorizations.