ThePhantomWolf
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Simplify the expression (3 – 3i)3 by performing operations with pure imaginary numbers and complex numbers. (Provide an explanation as I am terrible at calculus and need major assistance, thanks!)


A) –27 + 216i

B) –27 + 54i

C) –54 – 216i

D) –54 – 54i

Respuesta :

frika

Answer:

D

Step-by-step explanation:

Given: [tex]z=3-3i[/tex]

Find: [tex]z^3=(3-3i)^3[/tex]

Solution:

Use the formula

[tex](a-b)^3=a^3-3a^2b+3ab^2-b^3[/tex]

Hence,

[tex](3-3i)^3\\ \\=3^3-3\cdot 3^2\cdot(3i)+3\cdot 3\cdot (3i)^2-(3i)^3\\ \\=27-81i+9\cdot 9i^2-3^3i^3\\ \\=27-81i+81i^2-27i^3[/tex]

Now remind that

[tex]i^2=-1,[/tex]

then

[tex]i^3=i^2\cdot i=-1\cdot i=-i[/tex]

Substitute:

[tex](3-3i)^3\\ \\=27-81i+81\cdot (-1)-27\cdot (-i)\\ \\=27-81i-81+27i\\ \\=27-81-81i+27i\\ \\=-54-54i[/tex]

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