Question

If x = (ab) ^ (1/3) - (ab) ^ (- 1/3) prove that abx (x ^ 2 + 3) = a ^ 2 * b ^ 2 - 1​

Answer 1

Explanation:

x = (ab)^⅓ − (ab)^-⅓

Square both sides:

x² = (ab)^⅔ − 2 + (ab)^-⅔

Add 3 to both sides:

x² + 3 = (ab)^⅔ + 1 + (ab)^-⅔

Multiply both sides by x:

x (x² + 3) = x [(ab)^⅔ + 1 + (ab)^-⅔]

Substitute on the right hand side:

x (x² + 3) = ((ab)^⅓ − (ab)^-⅓) [(ab)^⅔ + 1 + (ab)^-⅔]

Distribute and simplify:

x (x² + 3) = (ab)^⅓ [(ab)^⅔ + 1 + (ab)^-⅔] − (ab)^-⅓ [(ab)^⅔ + 1 + (ab)^-⅔]

x (x² + 3) = ab + (ab)^⅓ + (ab)^-⅓ − [(ab)^⅓ + (ab)^-⅓ + (ab)⁻¹]

x (x² + 3) = ab + (ab)^⅓ + (ab)^-⅓ − (ab)^⅓ − (ab)^-⅓ − (ab)⁻¹

x (x² + 3) = ab − (ab)⁻¹

Multiply both sides by ab:

abx (x² + 3) = (ab)² − 1

abx (x² + 3) = a²b² − 1