Step 1: Understanding the Concept:
We are given an equation involving fractional powers. The goal is to manipulate the equation to find the value of a higher-degree polynomial expression. Cubing is a likely step to remove the fractional denominators. : Key Formula or Approach:
1. \( (a - b)^3 = a^3 - b^3 - 3ab(a - b) \).
2. Algebraic manipulation to match the target expression. Step 2: Detailed Explanation:
Given: \( x^{4/3} + x^{-1/3} = 1 \).
Multiply the entire equation by \( x^{1/3} \):
\[ x^{4/3} \cdot x^{1/3} + x^{-1/3} \cdot x^{1/3} = 1 \cdot x^{1/3} \]
\[ x + 1 = x^{1/3} \]
Rearranging terms:
\[ x^{5/3} - x^{1/3} = -1 \]
Cube both sides:
\[ (x^{5/3} - x^{1/3})^3 = (-1)^3 \]
Using the identity \( (a - b)^3 = a^3 - b^3 - 3ab(a - b) \):
\[ (x^{5/3})^3 - (x^{1/3})^3 - 3(x^{5/3})(x^{1/3})(x^{5/3} - x^{1/3}) = -1 \]
\[ x^5 - x - 3x^2(x^{5/3} - x^{1/3}) = -1 \]
Substitute \( (x^{5/3} - x^{1/3}) = -1 \) back into the equation:
\[ x^5 - x - 3x^2(-1) = -1 \]
\[ x^5 - x + 3x^2 = -1 \]. Step 3: Final Answer:
The value of \( x^5 + 3x^2 - x \) is -1.