Ответ:

x = 1; y = 2; z = 3

x = -1; y = -2; z = -3

Объяснение:

{x * y ^ 2 * z ^ 3 = 108 (1)

{x ^ 2 * y ^ 3 * z = 24 (3)

{x ^ 3 * y * z ^ 2 = 18 (2)

(1) : (2) =

= y * z / x ^ 2 = 6 (4)

(3) * (4) =

= y ^ 4 * z ^ 2 = 144 (5)

(1) : (3) =

= z ^ 2 / (x * y) = 4,5 (6)

(2) : (6) =

= x ^ 4 * y ^ 2 = 4 (7)

(3) : (2) =

= y ^ 2 / (x * z) = 4 / 3 (8)

(1) : (8) =

= x ^ 2 * z ^ 4 = 81 (9)

Теперь у нас есть система: (5), (7), (9).

{y ^ 4 * z ^ 2 = 144 (5)

{x ^ 4 * y ^ 2 = 4 (7)

{x ^ 2 * z ^ 4 = 81 (9)

(5) : (4) ^ 2 =

= z ^ 2 / x ^ 8 = 9 (10)

(9) : (10) ^ 2 =

= x ^ 18 = 1

(x ^ 9 - 1)(x ^ 9 + 1) = 0

(x ^ 3 - 1)(x ^ 6 + x ^ 3 + 1)(x ^ 3 + 1)(x ^ 6 - x ^ 3 + 1) = 0

(x - 1)(x ^ 2 + x + 1)(x + 1)(x ^ 2 - x + 1)(x ^ 6 + x ^ 3 + 1)(x ^ 6 - x ^ 3 + 1) = 0

1) x = 1

z ^ 2 = 9

(z - 3)(z + 3) = 0

1.1) z = 3

y ^ 4 = 16

(y ^ 2 - 4)(y ^ 2 + 4) = 0

(y - 2)(y + 2)(y ^ 2 + 4) = 0

y1 = 2

y2 = -2

1.2) z = -3

y ^ 4 = 16

(y ^ 2 - 4)(y ^ 2 + 4) = 0

(y - 2)(y + 2)(y ^ 2 + 4) = 0

y1 = 2

y2 = -2

2) x = -1

z ^ 2 = 9

(z - 3)(z + 3) = 0

2.1) z = 3

y ^ 4 = 16

(y ^ 2 - 4)(y ^ 2 + 4) = 0

(y - 2)(y + 2)(y ^ 2 + 4) = 0

y1 = 2

y2 = -2

2.2) z = -3

y ^ 4 = 16

(y ^ 2 - 4)(y ^ 2 + 4) = 0

(y - 2)(y + 2)(y ^ 2 + 4) = 0

y1 = 2

y2 = -2

У нас получились решения:

x = 1; y = 2; z = 3

x = 1; y = -2; z = 3

x = 1; y = 2; z = -3

x = 1; y = -2; z = -3

x = -1; y = 2; z = 3

x = -1; y = -2; z = 3

x = -1; y = 2; z = -3

x = -1; y = -2; z = -3

Сделав подстановку, получим:

x = 1; y = 2; z = 3

x = -1; y = -2; z = -3