Find the value of x^3 + y^3 + z^3 – 3xyz if x^2 + y^2 + z^2 = 83 and x + y + z = 15​

180

Step-by-step explanation:

Consider the equation x + y + z = 15

From algebraic identities, we know that (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)

So,

(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + xz)

From the question, x^2 + y^2 + z^2 = 83 and x + y + z = 15

So,

152 = 83 + 2(xy + yz + xz)

=> 225 – 83 = 2(xy + yz + xz)

Or, xy + yz + xz = 142/2 = 71

Using algebraic identity a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca),

x3 + y3 + z3 – 3xyz = (x + y + z)(x² + y² + z² – (xy + yz + xz))

Now,

x + y + z = 15, x² + y² + z² = 83 and xy + yz + xz = 71

So, x^3 + y^3 + z^3 – 3xyz = 15(83 – 71)

=> x^3 + y^3 + z^3 – 3xyz = 15 × 12

Or, x^3 + y^3 + z^3 – 3xyz = 180

Answered by GAUTHMATH

180

Step-by-step explanation:

1. 180°. the sum of the interior angles of a triangle is 180°, for the equilateral triangle that it has three equals side is 60°+60°+60°=180°.

2. 60°. each interior angle of a equilateral triangle is 60°

3. 120°. the measure of ∠m is 120°, the sum of the angles ∠m and ∠n is equal to 180°. so, in order to find the measure of ∠m, we have ∠m+∠n=180° > ∠m=180°-∠n > ∠m=180°-60°=120°.

4. 360°. the sum of the exterior angles of an equilateral triangle is ∠m+∠r+∠x=120°+120°+120°=360°. the exterior angles of any triangle always add 360°.