We are tasked with solving the equation:

\[
x^{\frac{2}{3}} + 4x = 12
\]

### Steps to solve:

1. **Rearrange the equation** to set it equal to 0:
   \[
   x^{\frac{2}{3}} + 4x - 12 = 0
   \]
2. **Substitute** a new variable to simplify the expression. Let:
   \[
   y = x^{\frac{1}{3}} \quad \text{so that} \quad y^2 = x^{\frac{2}{3}}
   \]
   The equation becomes:
   \[
   y^2 + 4y^3 - 12 = 0
   \]
3. **Solve this new equation** using substitution and algebraic methods (factoring or a numerical approach).

Would you like a detailed step-by-step breakdown or additional information on solving this equation?

Here are five follow-up questions to deepen your understanding:

1. How do you factor expressions involving fractional powers?
2. Can this equation have more than one real solution?
3. What are possible methods to solve cubic or fractional equations?
4. How do you check for extraneous solutions in such equations?
5. How does substitution simplify equations with powers and roots?

**Tip:** For equations with fractional exponents, using substitution is often the key to making them easier to handle!

This guide walks through solving the equation x^(2/3) + 4x = 12 using substitution. Learn key concepts like fractional exponents and the substitution method, with steps suitable for students in Grades 10-12.

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