Evaluate:$ \left(\frac{1}{2}\right)^{3}+\left(\frac{1}{3}\right)^{3}-\left(\frac{5}{6}\right)^{3} $

Given: 

\( \left(\frac{1}{2}\right)^{3}+\left(\frac{1}{3}\right)^{3}-\left(\frac{5}{6}\right)^{3} \)

To do: 

We have to evaluate the given expression.

Solution: 

We know that,

If $x+y+z=0$, then $x^{3}+y^{3}+z^{3}=3 x y z$.

Let $a=\frac{1}{2}, b=\frac{1}{3}$ and $c=\frac{-5}{6}$

$a+b+c=\frac{1}{2}+\frac{1}{3}-\frac{5}{6}$

$=\frac{3+2-5}{6}$

$=\frac{0}{6}$

$=0$

This implies,

$a^{3}+b^{3}+c^{3}=3 a b c$

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

$=-3 \times \frac{1}{2} \times \frac{1}{3} \times \frac{5}{6}$

$=\frac{-5}{12}$

Hence, $(\frac{1}{2})^{3}+(\frac{1}{3})^{3}-(\frac{5}{6})^{3}=\frac{-5}{12}$.

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Updated on: 10-Oct-2022

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