[tex]2-11i \text{ is the standard form of given expression }[/tex]
Solution:
The standard form of complex number is: a + bi
where a is the real part and bi is the imaginary part
Given expression is:
[tex](2-i)^3[/tex]
Expand the above expression using algebraic identity
[tex](a-b)^3=a^3-b^3-3ab(a-b)[/tex]
[tex]\text{For } (2-i)^3 \text{ we get, a = 2 and b = i}[/tex]
Thus on expanding using the above algebraic identity we get,
[tex](2-i)^3=(2)^3-(i)^3-3(2)(i)(2-i)[/tex]
Simplify the above expression
[tex](2-i)^3=8 -i^3-6i(2-i)\\\\(2-i)^3=8 -i^3-12i+6i^2[/tex]
We know that,
[tex]i^2 = -1\\\\i^3 = -i[/tex]
Substituting in above simplified expression, we get,
[tex](2-i)^3=8-(-i)-12i+6(-1)\\\\(2-i)^3=8 + i -12i -6\\\\\text{Combine the like terms }\\\\(2-i)^3=8 - 6 + i -12i\\\\(2-i)^3=2-11i[/tex]
Thus the given expression is expressed in standard form
