Question

without actual calculating the cubes, find the value of (½)³+(⅓)³-(5/6)³​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

If x + y + z = 0 then

${x}^{3} + {y}^{3} + {z}^{3} = 3xyz$

TO EVALUATE

$\: \displaystyle \: { \bigg( \frac{1}{2} \bigg)}^{3} + { \bigg( \frac{1}{3} \bigg)}^{3} - { \bigg( \frac{5}{6} \bigg)}^{3}$

EVALUATION

Let

$\displaystyle \: x = \frac{1}{2} \: , \: y = \frac{1}{3} \: , \: z - \frac{5}{6}$

Then

$\displaystyle \: x + y + z = \frac{1}{2} + \frac{1}{3} - \frac{5}{6} = \frac{(3 + 2 - 5)}{6} = 0$

So by the above mentioned formula we get

$\displaystyle \: {x}^{3} + {y}^{3} + {z}^{3} = 3xyz$

$\implies \: \displaystyle \: { \bigg( \frac{1}{2} \bigg)}^{3} + { \bigg( \frac{1}{3} \bigg)}^{3} + { \bigg( - \frac{5}{6} \bigg)}^{3} = 3 \times \: \bigg( \frac{1}{2} \bigg) \times \bigg( \frac{1}{3} \bigg) \times \bigg( - \frac{5}{6} \bigg)$

$\implies \: \displaystyle \: { \bigg( \frac{1}{2} \bigg)}^{3} + { \bigg( \frac{1}{3} \bigg)}^{3} + { \bigg( - \frac{5}{6} \bigg)}^{3} = - \: \frac{5}{12}$

RESULT

$\boxed{ \: \: \displaystyle \: { \bigg( \frac{1}{2} \bigg)}^{3} + { \bigg( \frac{1}{3} \bigg)}^{3} - { \bigg( \frac{5}{6} \bigg)}^{3} = - \: \frac{5}{12} \: }$