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.