Evaluate:
\[ \left(\frac12\right)^3 + \left(\frac13\right)^3 – \left(\frac56\right)^3 \]
Solution:
Rewrite the expression:
\[ \left(\frac12\right)^3 + \left(\frac13\right)^3 + \left(-\frac56\right)^3 \]
Since, \[ \frac12+\frac13-\frac56=0 \]
Using identity:
\[ a^3+b^3+c^3=3abc \quad \text{when} \quad a+b+c=0 \]
Take \[ a=\frac12,\qquad b=\frac13,\qquad c=-\frac56 \]
\[ \frac12+\frac13-\frac56=0 \]
Therefore, \[ \left(\frac12\right)^3 + \left(\frac13\right)^3 – \left(\frac56\right)^3 = 3\left(\frac12\right)\left(\frac13\right)\left(-\frac56\right) \]
\[ = -\frac{15}{36} \]
\[ = -\frac{5}{12} \]