Question

if a+b+c = 12, a*+b*+c*=90 , then find the value of a cube+ b cube+ c cube- 3abc.

Answer 1

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Answer 2

a + b + c = 12

a² + b² + c² = 90

For finding the value of a³ + b³ + c³ - 3abc, we have to find the value of ab + bc + ac

in a + b + c, squaring both sides

(a + b + c)² = (12)²

using the identity,

a² + b² + c² + 2ab + 2bc + 2ac = 144

putting the value of a² + b² + c² as given,

90 + 2ab + 2bc + 2ac = 144

=> 2ab + 2bc + 2ac = 54

Taking 2 common,

2( ab + bc + ac) = 54

=> ab + bc + ac = 54/2

=> ab + bc + ac = 27


Now using the identity

a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ac)
we can find the required answer,

So,
a³ + b³ + c³ - 3abc = (12)(90 - 27)

=> a³ + b³ + c³ - 3abc = 12 × 63

=> a³ + b³ + c³ - 3abc = 756


Hope it helps dear friend ☺️✌️