Question

Evaluate: \[ \frac{5}{\sqrt[3]{16}-\sqrt[3]{4}+1}-\frac{3}{\sqrt[3]{16}+\sqrt[3]{4}+1} \]

• A. \(\frac14\)
• B. \(\frac13\)
• C. \(2\)
• D. \(\frac12\)

Hint:

For cube-root expressions, substitution and identities like \(a^3 \pm b^3\) simplify calculations quickly.

Answer 1

The Correct Option is C

Concept: Use substitution to simplify cube roots.

Step 1: Substitute the cube root values.
Let: \[ x=\sqrt[3]{4} \] Then: \[ \sqrt[3]{16}=\sqrt[3]{4^2}=x^2 \] Given expression becomes: \[ \frac{5}{x^2-x+1}-\frac{3}{x^2+x+1} \]

Step 2: Use the identity.
Since: \[ x^3=4 \] we know: \[ x^3+1=5 \] Factorizing: \[ x^3+1=(x+1)(x^2-x+1) \] So: \[ 5=(x+1)(x^2-x+1) \] Thus: \[ \frac{5}{x^2-x+1}=x+1 \] Also: \[ x^3-1=3 \] Factorizing: \[ x^3-1=(x-1)(x^2+x+1) \] So: \[ 3=(x-1)(x^2+x+1) \] Thus: \[ \frac{3}{x^2+x+1}=x-1 \]

Step 3: Substitute back.
\[ (x+1)-(x-1) \] \[ =x+1-x+1 \] \[ =2 \] Thus, the required value is: \[ \boxed{2} \]