Question

How do you simplify \[{{\left( 6i \right)}^{3}}\]?

Answer 1

Hint: Assume the value of the given expression as ‘E’. Apply the formula of the topic ‘Exponents and powers’ given as: - \[{{\left( ab \right)}^{m}}={{a}^{m}}\times {{b}^{m}}\], to simplify the given expression. Here, ‘a’ and ‘b’ are called the bases and ‘m’ is the exponent. Use the relation: - \[{{i}^{2}}=-1\], where ‘i’ is the imaginary number \[\sqrt{-1}\], for further simplification of ‘E’ and get the answer.

Complete step by step answer:
Here, we have been provided with the expression \[{{\left( 6i \right)}^{3}}\] and we have been asked to simplify it.
Now, let us assume the given expression as ‘E’. So, we have,
\[\Rightarrow E={{\left( 6i \right)}^{3}}\]
Applying the formula: - \[{{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}\], where ‘a’ and ‘b’ are called the basses and ‘m’ is the exponent, so we get,
\[\begin{align}
  & \Rightarrow E={{6}^{3}}\times {{i}^{3}} \\
 & \Rightarrow E=216\times {{i}^{3}} \\
\end{align}\]
Now, here we can see that in the above expression we have an alphabet ‘i’, actually it is the notation for the imaginary number \[\sqrt{-1}\]. ‘i’ is the solution of the quadratic equation \[{{x}^{2}}+1=0\]. There are no real solutions of this quadratic equation and therefore the concept of imaginary numbers and complex numbers arises. A complex number is written in general form as: - \[z=a+ib\], where ‘z’ is the notation of complex numbers, ‘a’ is the real part and ‘ib’ is the imaginary part. Here, \[i=\sqrt{-1}\].
Now, let us come back to the expression ‘E’. We can write the obtained expression as: - \[\Rightarrow E=216\times {{i}^{2}}\times i\] - (1)
Since, \[i=\sqrt{-1}\], therefore on squaring both the sides, we get,
\[\Rightarrow {{i}^{2}}=-1\]
So, substituting the value of \[{{i}^{2}}\] in equation (1), we get,
\[\begin{align}
  & \Rightarrow E=216\times \left( -1 \right)\times i \\
 & \Rightarrow E=-216i \\
\end{align}\]
Hence, the above expression is the simplified form and our answer.

Note:
One must not consider ‘i’ as any variable or just an alphabet. Remember that ‘i’ always denotes the imaginary number \[\sqrt{-1}\] in the topic ‘complex numbers’. You must remember the formulas of the topic ‘exponents and power’ like: - \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}},{{a}^{m}}\div {{a}^{n}}={{a}^{m-n}},{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], because these formulas are frequently used in other topics of mathematics. Remember the concepts of complex numbers and their general forms.