Question

Find the value of x3 + y3 + z3 – 3xyz if x2 + y2 + z2 = 83 and x + y + z = 15​

Answer 1

$\bf\underline{Question:-}$

→ Find the value of x³+ y³ + z³ – 3xyz

       if x² + y² + z² = 83

        and x + y + z = 15​

____________________

$\bf\underline{Answer:-}$

Given :

x² + y² + z² = 83

And

x + y + z = 15

To Find :

The value of ,  x³ + y³ + z³ - 3 x y z

Solution :

∵  ( x + y + z )² = x² + y² + z² + 2 ( x y + y z + z x )

And

x + y + z = 15

So,  83 + 2 ( x y + y z + z x ) = ( 15 )²

Or, 2 ( x y + y z + z x ) = 225 - 83

Or, 2 ( x y + y z + z x ) = 142

∴ ( x y + y z + z x ) =  $\frac{142}{2}$

i.e   ( x y + y z + z x ) = 71

Again  :

∵  x³ + y³ + z³ - 3 x y z = ( x + y + z ) [ ( x² + y² + z² ) - ( x y + y z + z x ) ]

Or,                                = ( 15 ) × [ 83 - 71 ]

Or,                                 = 15 × 12

that is                                = 180

Hence, The value of x³ + y³ + z³ - 3 x y z is 180 .

Answer 2

Answer:

\bf\underline{Question:-}Question:−

→ Find the value of x³+ y³ + z³ – 3xyz

       if x² + y² + z² = 83

        and x + y + z = 15

____________________

\bf\underline{Answer:-}Answer:−

Given :

x² + y² + z² = 83

And

x + y + z = 15

To Find :

The value of ,  x³ + y³ + z³ - 3 x y z

Solution :

∵  ( x + y + z )² = x² + y² + z² + 2 ( x y + y z + z x )

And

x + y + z = 15

So,  83 + 2 ( x y + y z + z x ) = ( 15 )²

Or, 2 ( x y + y z + z x ) = 225 - 83

Or, 2 ( x y + y z + z x ) = 142

∴ ( x y + y z + z x ) =  \frac{142}{2}2142

i.e   ( x y + y z + z x ) = 71

Again  :

∵  x³ + y³ + z³ - 3 x y z = ( x + y + z ) [ ( x² + y² + z² ) - ( x y + y z + z x ) ]

Or,                                = ( 15 ) × [ 83 - 71 ]

Or,                                 = 15 × 12

that is                                = 180

Hence, The value of x³ + y³ + z³ - 3 x y z is 180 .