Question

x+y+z=3 x²+y²+z²=5 x³+y³+z³=7 Find x⁴+y⁴+z⁴ First one to answer will be marked brainless.

Answer 1

Given:

x + y + z = 3                            (i)

x² + y² + z² = 5                       (ii)

x³ + y³ + z³ = 7                       (iii)

To find:

x⁴ + y⁴ + z⁴ = ?

Solution:

Take square of equation (i)

(x + y + z)² = 3²

x² + y² + z² + 2(xy + yz + xz) = 9

From equation (ii) we can put the value

5 + 2(xy + yz + xz) = 9

2(xy + yz + xz) = 5 - 9

2(xy + yz + xz) = 4

xy + yz + xz = 2                                                        (iv)

Take square of equation (iv)

(xy + yz + xz)² = 2²

x²y² + y²z² + x²z² + 2xyz(x + y + z) = 4

From equation (i) we can put the value

x²y² + y²z² + x²z² + 2xyz(3) = 4

x²y² + y²z² + x²z² + 6xyz = 4                                    (v)

We know that x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - xz)

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

From equation (i), (ii), (iii), and (iv) put value in the above equation

7 - 3xyz = 3(5 - 2)

7 - 3xyz = 3 × 3

3xyz = 7 - 9

xyz = -2/3

Put the value in equation (v)

x²y² + y²z² + x²z² + 6(-2/3) = 4

x²y² + y²z² + x²z² - 4 = 4

x²y² + y²z² + x²z² = 8                                                   (vi)

Take the square of equation (ii)

(x² + y² + z²)² = 5²

x⁴ + y⁴ + z⁴ + 2(x²y² + y²z² + x²z²) = 25                        

Put value from equation (vi)

x⁴ + y⁴ + z⁴ + 2(8) = 25

x⁴ + y⁴ + z⁴ + 16 = 25

x⁴ + y⁴ + z⁴ = 25 - 16

x⁴ + y⁴ + z⁴ = 9

Answer 2

Answer:

Step-by-step explanation: