Question

Find the value of the complex expression: \[{\left( {1 + i} \right)^6}\].
A. \[ - 8i\]
B. \[8i\]
C. Does not exist
D. Cannot be determined

Answer 1

Hint: We will apply the expansion formulae for ‘\[{\left( {a + b} \right)^2}\]’ by adjusting the given power of the expression by using the indices rule that is ‘\[{\left( {{a^m}} \right)^n} = {a^{mn}}\]’ respectively. Hence, using the definition of the complex part that is ‘\[i = \sqrt { - 1} \]’, the desired solution/value is obtained.

Complete step-by-step answer:
Given, \[{\left( {1 + i} \right)^6}\] is the complex expression as it seems an imaginary part ‘\[i\]’ which the mathematical value resembles to be the ‘\[\sqrt { - 1} \]’ respectively.
Hence, solving it in accordance to complex relations or the certain formulae by using them, we can solve the desire expression
Hence, first of all adjusting its power that is ‘\[6\]’ so that we can use the algebraic identity that is ‘\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]’, by using the certain rules of indices for ‘\[{\left( {{a^m}} \right)^n} = {a^{mn}}\]’ respectively, we get
\[ \Rightarrow {\left( {1 + i} \right)^6} = {\left[ {{{\left( {1 + i} \right)}^2}} \right]^3}\]
Now,
Hence, expanding the certain algebraic identity, we get
\[ \Rightarrow {\left( {1 + i} \right)^6} = {\left( {1 + 2i + {i^2}} \right)^3}\] … (i)
Since, by the definition of complex identity ‘\[i = \sqrt { - 1} \]’
Hence, we can find the remaining parameters/factors of required terms that is
If \[i = \sqrt { - 1} \] then,
\[{i^2} = - 1\],
\[{i^3} = - i\],
\[{i^4} = 1\], and so on.
Hence, equation (i) becomes
\[ \Rightarrow {\left( {1 + i} \right)^6} = {\left( {1 + 2i - 1} \right)^3}\] … (\[\because {i^2} = - 1\])
Solving the equation mathematically, we get
\[ \Rightarrow {\left( {1 + i} \right)^6} = {\left( {2i} \right)^3}\]
As a result, cubing the certain terms, we get
\[ \Rightarrow {\left( {1 + i} \right)^6} = 8{i^3}\]
\[ \Rightarrow {\left( {1 + i} \right)^6} = - 8i\] … (\[\because {i^3} = - i\])
Therefore, the value of the complex expression \[{\left( {1 + i} \right)^6} = -8i\]. So, Option (A) is correct.

Note: One must be able to remember the complex relation for ‘\[i\]’ such as ‘\[{i^2} = - 1\]’, ‘\[{i^3} = - i\]’, ‘\[{i^4} = 1\]’, ‘\[{i^5} = i\]’, and so on. Also, the various algebraic expansion formulae (or, identity); indices rules that seems like ‘\[{\left( {a + b} \right)^2}\]’, ‘\[{\left( {a + b} \right)^3}\]’, ‘\[\left( {{a^2} - {b^2}} \right)\]’, ‘\[{\left( {a - b} \right)^2}\]’, ‘\[{\left( {a - b} \right)^3}\]’, ‘\[{a^m} \times {a^n} = {a^{m + n}}\]’, ‘\[{\left( {{a^m}} \right)^n} = {a^{mn}}\]’, ‘\[\dfrac{1}{{{a^m}}} = {a^{ - m}}\]’, etc. so as to be sure of the final answer.