Write the Value of (1/2)³ + (1/3)³ − (5/6)³
Write the value of \[ \left(\frac{1}{2}\right)^3 + \left(\frac{1}{3}\right)^3 – \left(\frac{5}{6}\right)^3 \]
Solution
Let
\[ a=\frac{1}{2},\quad b=\frac{1}{3},\quad c=-\frac{5}{6} \]
Then,
\[ a+b+c= \frac{1}{2} + \frac{1}{3} – \frac{5}{6} \]
Taking LCM = 6,
\[ a+b+c= \frac{3+2-5}{6} \]
\[ a+b+c=0 \]
We know that if \(a+b+c=0\), then
\[ a^3+b^3+c^3=3abc \]
Therefore,
\[ \left(\frac{1}{2}\right)^3 + \left(\frac{1}{3}\right)^3 – \left(\frac{5}{6}\right)^3 = 3\left(\frac{1}{2}\right) \left(\frac{1}{3}\right) \left(-\frac{5}{6}\right) \]
\[ = -\frac{5}{12} \]
Therefore,
\[ \boxed{-\frac{5}{12}} \]