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