Using Rule 8,
x^{1 / 3} + y^{1 / 3} = z^{1 / 3} ......(i)
Cubing both sides,
( x^{1 / 3} + y^{1 / 3} = z^{1 / 3} )³ = z
⇒ x + y + 3 x^{1 / 3} . y^{1 / 3} ( ^{1 / 3} + y^{1 / 3} ) = z
[∴ (a + b)³ = a³ + b³ + 3ab (a + b)]
⇒ x + y – z = - 3 x^{1 / 3} . y^{1 / 3} .z^{1 / 3} ......(ii) [From equation (i)]
∴ (x + y – z)³ + 27xyz = [ - 3x^{1 / 3} . y^{1 / 3} . z^{1 / 3} ]³ + 27xyz [From equation (ii)]
= – 27xyz + 27xyz = 0

Correct Option: C

Using Rule 8,
x^{1 / 3} + y^{1 / 3} = z^{1 / 3} ......(i)
Cubing both sides,
( x^{1 / 3} + y^{1 / 3} = z^{1 / 3} )³ = z
⇒ x + y + 3 x^{1 / 3} . y^{1 / 3} ( ^{1 / 3} + y^{1 / 3} ) = z
[∴ (a + b)³ = a³ + b³ + 3ab (a + b)]
⇒ x + y – z = - 3 x^{1 / 3} . y^{1 / 3} .z^{1 / 3} ......(ii) [From equation (i)]
∴ (x + y – z)³ + 27xyz = [ - 3x^{1 / 3} . y^{1 / 3} . z^{1 / 3} ]³ + 27xyz [From equation (ii)]
= – 27xyz + 27xyz = 0

Using Rule 8,
Given, x + y = 7
Now, x³ + y³ + 21xy = (x + y)³ – 3xy(x + y) + 21xy
= (7)³ – 3xy(7) + 21xy
= 343 – 21xy + 21xy = 343

Correct Option: C

Using Rule 8,
Given, x + y = 7
Now, x³ + y³ + 21xy = (x + y)³ – 3xy(x + y) + 21xy
= (7)³ – 3xy(7) + 21xy
= 343 – 21xy + 21xy = 343

Using Rule 8,
x = √3 + √2

x3 +

1

=

x +

1

3

- 3x .

1

x +

1

x3

x

x

x

Correct Option: C

Using Rule 8,
x = √3 + √2

x3 +

1

=

x +

1

3

- 3x .

1

x +

1

x3

x

x

x

Using Rule 21,
x + y + z = 6
⇒ x + y + z – 6 = 0
⇒ (x – 1) + (y – 2) + (z – 3) = 0
If a + b + c = 0, then a³ + b³ + c³ = 3abc
∴ (x – 1)³ + (y – 2)³ + (z – 3)³ = 3 (x – 1) (y – 2) (z – 3)

Correct Option: D

Using Rule 21,
x + y + z = 6
⇒ x + y + z – 6 = 0
⇒ (x – 1) + (y – 2) + (z – 3) = 0
If a + b + c = 0, then a³ + b³ + c³ = 3abc
∴ (x – 1)³ + (y – 2)³ + (z – 3)³ = 3 (x – 1) (y – 2) (z – 3)

x + y + z = 6
On squaring,
x² + y² + z² + 2xy + 2zy + 2zx = 36
⇒ 20 + 2 (xy + yz + zx) = 36
⇒ xy + yz + zx = 8
∴ x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx)
= 6 (20 – 8)
= 72

Correct Option: C

x + y + z = 6
On squaring,
x² + y² + z² + 2xy + 2zy + 2zx = 36
⇒ 20 + 2 (xy + yz + zx) = 36
⇒ xy + yz + zx = 8
∴ x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx)
= 6 (20 – 8)
= 72