Question

The solutions of \( x^{2/5} + 3x^{1/5} - 4 = 0 \) are

• A. \( 1,1024 \)
• B. \( -1,1024 \)
• C. \( 1,-1024 \)
• D. \( -1024,1 \)
• E. \( -1,1031 \)

Hint:

Always substitute to reduce fractional powers.

Answer 1

The Correct Option is C

Concept: Substitution simplifies fractional exponents.

Step 1: Let \( y = x^{1/5} \).
Then: \[ x^{2/5} = y^2 \] Equation becomes: \[ y^2 + 3y - 4 = 0 \]

Step 2: Solve quadratic equation.
\[ y^2 + 3y - 4 = (y+4)(y-1) = 0 \] \[ y = 1 \quad \text{or} \quad y = -4 \]

Step 3: Back substitute.
\[ x^{1/5} = 1 \Rightarrow x = 1 \] \[ x^{1/5} = -4 \Rightarrow x = (-4)^5 = -1024 \]

Step 4: Verify both solutions satisfy equation.
Substitute values → both valid.

Step 5: Final solution set.
\[ x = 1, -1024 \]