Question

Write the value of: (1/2)³ + (1/3)³ - (5/6)³

Answer 1

Given : (½)³ + (⅓)³ - (⅚)³

Let x = ½ , y = ⅓  & z = - ⅚  

x + y + z = ½  +(⅓ ) + (-⅚ ) = ⅚ - ⅚ = 0

Here, x + y + z = 0

We know that if,   x + y + z = 0, then x³ + y³ + z³ = 3xyz

(½ )³ + (⅓ )³ + (-⅚ )³ = 3(½ )(⅓ )(-⅚ ) = - 5/12

(½ )³ + (⅓ )³ + (-⅚ )³ = - 5/12

Hence, the value of (1/2)³ + (1/3)³ - (5/6)³ is - 5/12.

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Answer 2

The identity states:

a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ac)

Here, a = 1/2

b = 1/3

c = -5/6

a + b + c = 1/2 + 1/3 - 5/6 = 0

Thus, a + b + c = 0

So, a³ + b³ + c³ = 3abc = 3(1/3)(1/2)(-5/6)

= -5/12