Question
Write the value of $30^3+20^3-50^3$.
✓
Answer
The given expression is
$30^3+20^3-50^3$
Let $a=30, b=20$ and $c=-50$. Then the given expression becomes $30^3+20^3-50^3=a^3+b^3+c^3$
Note that
$a+b+c=30+20+(-50)$
$=30+20-50$
$=0$
Recall the formula
$a^3+b^3+c^3-3 a b c=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
when $a+b+c=0$, this becomes
$a^3+b^3+c^3-3 a b c=0 \cdot\left(a^2+b^2+c^2-a b-b c-c a\right)$
$=0$
$a^3+b^3+c^3=3 a b c$
So, we have the new formula
$a^3+b^3+c^3=3 a b c \text {, when } a+b+c=0$
Using the above formula, the value of the given expression is
$a^3+b^3+c^3=3 a b c$
$30^3+20^3-50^3=3 \cdot(30) \cdot(20) \cdot(-50)$
$30^3+20^3-50^3=-90000$
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