Question

(x⅓ - x-⅓)(x⅔ + 1 + x-⅔)​

Answer 1

Answer:

Zero

Explanation:

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Answer 2

Answer:

x = 1, x =-8

Step-by-step

The given equation

x⅔ + x⅓ - 2 = 0

can be written as (x⅓)² + x⅓ - 2 = 0 ………………………………………………….(1)

If y = x⅓ then (x⅓)² = y² and (1) →

y² + y - 2 = 0

Or, (y² - 1) + y - 1 = 0

Factorising the quantity within brackets,

(y+1) (y-1) + (y-1) = 0

Or, (y-1) (y+1 + 1) =0 [the common factor y-1 is taken out]

Or, (y - 1) (y + 2) = 0

Or, y - 1 = 0, y + 2 = 0

⇒ y = 1 , y = -2

⇒ x⅓ = 1, x⅓ = -2 (∵ y = x⅓)

Cubing (x⅓)³ = 1³ , (x⅓)³ = (-2)³

⇒ x = 1, x = -8 (Answer)

Verification:

x = 1

L.H.S. of (1) = (x⅓)² + x⅓ - 2 = (1⅓)² + 1⅓ - 2 = 1²+ 1 - 2 = 1² + 1 - 2 = 1+1 = 0 = R.H.S.

x = -8

L.H.S. of (1) = (-8⅓)² + (-8)⅓ - 2 = (-2)² - 2 - 2 = 4 - 2 - 2 = 4 - 4 = 0 = R.H.S.

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