How do you simplify ${{i}^{59}}$?

Hint: To solve this question, we need to make the exponent on $i$ equal to a multiple of four. Since the exponent is equal to $59$ which is one less than $60$, a multiple of four, we need to multiply and divide the given expression by $i$ to obtain $\dfrac{{{i}^{60}}}{i}$. Then, we have to use the relation ${{i}^{4}}=1$ to show that any multiple of four, raised to $i$ given one. Using this our expression will get reduced to $\dfrac{1}{i}$. Finally, on dividing and multiplying the obtained expression by $i$ we will get the simplified expression.

Complete step by step solution:
Let us write the expression given in the above question in the below equation as
$\Rightarrow E={{i}^{59}}........\left( i \right)$
Now, we know that $i$ is equal to the square root one minus one, which in turn means that the square of $i$ is equal to minus one, that is,
\[\Rightarrow {{i}^{2}}=-1.......\left( ii \right)\]
Squaring both the sides, we get
$\begin{align}
  & \Rightarrow {{\left( {{i}^{2}} \right)}^{2}}={{\left( -1 \right)}^{2}} \\
 & \Rightarrow {{i}^{4}}=1 \\
\end{align}$
Raising the terms on both sides of the above equation to the exponent of $n$, where $n$ is a natural number, we get
$\begin{align}
  & \Rightarrow {{\left( {{i}^{4}} \right)}^{n}}={{\left( 1 \right)}^{n}} \\
 & \Rightarrow {{i}^{4n}}=1 \\
\end{align}$
From the above equation, we can say that the value of $i$ raised to a multiple of four is equal to one.
Now, we consider the equation (i)
$\Rightarrow E={{i}^{59}}$
Multiplying and dividing by $i$ we get
$\begin{align}
  & \Rightarrow E={{i}^{59}}\times \dfrac{i}{i} \\
 & \Rightarrow E=\dfrac{{{i}^{60}}}{i} \\
\end{align}$
Since $60$ is a multiple of four, we can substitute ${{i}^{60}}=1$ in the above equation to get
$\Rightarrow E=\dfrac{1}{i}$
Multiplying and dividing by $i$ we get
\[\Rightarrow E=\dfrac{i}{{{i}^{2}}}\]
Finally, substituting (ii) in the above equation, we get
$\begin{align}
  & \Rightarrow E=\dfrac{i}{-1} \\
 & \Rightarrow E=-i \\
\end{align}$

Hence, the given expression is simplified as $-i$.

Note: The equation ${{i}^{4n}}=1$, which we obtained in the above solution is an identity. We must remember it in order to quickly solve these kinds of problems. Also, we must remember the identities \[{{i}^{2}}=-1\] and $\dfrac{1}{i}=-i$ must be remembered for solving these types of questions.