Question

Find the value of (2+i+i²)³​

Answer 1

Answer:

The value of ( 2 + i + i² )³ is - 2 + 2i.

Step-by-step-explanation:

The given complex number is ( 2 + i + i² )³.

We have to evaluate the given complex number.

Now, we know that,

( a + b + c )³ = a³ + b³ + c³ + 3 ( a + b ) ( b + c ) ( a + c )

By using this identity,

( 2 + i + i² )³

⇒ 2³ + i³ + ( i² )³ + 3 ( 2 + i ) ( i + i² ) ( 2 + i² )

We know that,

• i² = - 1
• i³ = - i

Now,

⇒ 8 + ( - i ) + ( - 1 )³ + 3 ( 2 + i ) [ i + ( - 1 ) ] [ 2 + ( - 1 ) ]

⇒ 8 - i - 1 + 3 ( 2 + i ) ( i - 1 ) ( 2 - 1 )

⇒ 8 - 1 - i + 3 ( 2 + i ) ( i - 1 ) * 1

⇒ 7 - i + ( 6 + 3i ) ( i - 1 )

⇒ 7 - i + 6 ( i - 1 ) + 3i ( i - 1 )

⇒ 7 - i + 6i - 6 + 3i² - 3i

⇒ 7 - 6 - i + 6i - 3i + 3 * ( - 1 )

⇒ 1 + 5i - 3i - 3

⇒ 1 - 3 + 5i - 3i

⇒ - 2 + 2i

∴ The value of ( 2 + i + i² )³ is - 2 + 2i.

Answer 2

Answer:

The value of(2+i+i²)³ is -2+2i