Question

if a+b+c=15,a²+b²+c²=83.find the value of a³+b³+c³-3abc​

Answer 1

Answer:

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Answer 2

The value of a³ + b³ + c³ - 3abc is 180.

Given :

• a + b + c = 15
• a² + b² + c² = 83

To Find :

The value of a³ + b³ + c³ - 3abc

Solution :

As we know,

$\bigstar \: \boxed{\sf a^3 + b^3 + c^3 - 3abc = (a + b + c) (a^2 + b^2 + c^2- ab - bc - ca)}$

So, Let's consider it as Equation 1,

Now,

★ (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) ★

→ 15² = 83 + 2(ab + bc + ca)

→ 225 = 83 + 2(ab + bc + ca)

→ 225 - 83 = 2(ab + bc + ca)

→ 142 = 2(ab + bc + ca)

→ ab + bc + ca = 71

Now, Let's put the value of ab + bc + ca in Equation 1,

★ a³ + b³ + c³ - 3abc = (a + b + c) {(a² + b² + c²) - ab + bc + ca)} ★

→ a³ + b³ + c³ - 3abc = 15 × (83 - 71)

→ a³ + b³ + c³ - 3abc = 15 × 12

→ a³ + b³ + c³ - 3abc = 180

∴ The value of a³ + b³ + c³ - 3abc is 180.