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Question

Question: How do you simplify \({{i}^{59}}\)?...

How do you simplify i59{{i}^{59}}?

Explanation

Solution

To solve this question, we need to make the exponent on ii equal to a multiple of four. Since the exponent is equal to 5959 which is one less than 6060, a multiple of four, we need to multiply and divide the given expression by ii to obtain i60i\dfrac{{{i}^{60}}}{i}. Then, we have to use the relation i4=1{{i}^{4}}=1 to show that any multiple of four, raised to ii given one. Using this our expression will get reduced to 1i\dfrac{1}{i}. Finally, on dividing and multiplying the obtained expression by ii 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
E=i59........(i)\Rightarrow E={{i}^{59}}........\left( i \right)
Now, we know that ii is equal to the square root one minus one, which in turn means that the square of ii is equal to minus one, that is,
i2=1.......(ii)\Rightarrow {{i}^{2}}=-1.......\left( ii \right)
Squaring both the sides, we get
(i2)2=(1)2 i4=1 \begin{aligned} & \Rightarrow {{\left( {{i}^{2}} \right)}^{2}}={{\left( -1 \right)}^{2}} \\\ & \Rightarrow {{i}^{4}}=1 \\\ \end{aligned}
Raising the terms on both sides of the above equation to the exponent of nn, where nn is a natural number, we get
(i4)n=(1)n i4n=1 \begin{aligned} & \Rightarrow {{\left( {{i}^{4}} \right)}^{n}}={{\left( 1 \right)}^{n}} \\\ & \Rightarrow {{i}^{4n}}=1 \\\ \end{aligned}
From the above equation, we can say that the value of ii raised to a multiple of four is equal to one.
Now, we consider the equation (i)
E=i59\Rightarrow E={{i}^{59}}
Multiplying and dividing by ii we get
E=i59×ii E=i60i \begin{aligned} & \Rightarrow E={{i}^{59}}\times \dfrac{i}{i} \\\ & \Rightarrow E=\dfrac{{{i}^{60}}}{i} \\\ \end{aligned}
Since 6060 is a multiple of four, we can substitute i60=1{{i}^{60}}=1 in the above equation to get
E=1i\Rightarrow E=\dfrac{1}{i}
Multiplying and dividing by ii we get
E=ii2\Rightarrow E=\dfrac{i}{{{i}^{2}}}
Finally, substituting (ii) in the above equation, we get
E=i1 E=i \begin{aligned} & \Rightarrow E=\dfrac{i}{-1} \\\ & \Rightarrow E=-i \\\ \end{aligned}

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

Note: The equation i4n=1{{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 i2=1{{i}^{2}}=-1 and 1i=i\dfrac{1}{i}=-i must be remembered for solving these types of questions.