To factor the expression $x^{\frac{1}{2}} - x^{\frac{1}{4}} - 30$, we can use a substitution method to simplify it.

Step-by-step process:

1. Substitute a new variable: Let $y = x^{\frac{1}{4}}$. This means: $y^2 = x^{\frac{1}{2}}.$ So the expression becomes: $y^2 - y - 30.$

2. Factor the quadratic expression: Now, we factor the quadratic: $y^2 - y - 30 = (y - 6)(y + 5).$

3. Substitute back for $y$: Since $y = x^{\frac{1}{4}}$, substitute it back: $(x^{\frac{1}{4}} - 6)(x^{\frac{1}{4}} + 5).$

Thus, the factored form of the expression is: $(x^{\frac{1}{4}} - 6)(x^{\frac{1}{4}} + 5).$

Would you like more details on any of the steps?

Related Questions:

1. How can you solve equations involving fractional exponents?
2. What is the method for solving quadratic equations by factoring?
3. How can substitution simplify complex algebraic expressions?
4. What are other common techniques to factor quadratic expressions?
5. How do fractional exponents relate to roots of numbers?

Tip: Substitution is a powerful tool in algebra when dealing with complex expressions. Always look for patterns that can be simplified!