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Question

Question: How do you simplify \(\dfrac{{4 - 6i}}{i}\)?...

How do you simplify 46ii\dfrac{{4 - 6i}}{i}?

Explanation

Solution

The question belongs to the simplification of complex numbers. Any fraction of complex numbers can be represented in only (a+ib)\left( {a + ib} \right) form by doing some basic arithmetic operations. We know that the value of $${i^2} = - 1inthecomplexnumbers.Wecanreplaceminussignsbysimplyreplacingitwithin the complex numbers. We can replace minus signs by simply replacing it with i2{i^2} $. To convert the given fraction into the simplest form, first we will look for the numerator of the fraction and we will write it into the simplest form by taking a common factor of it. We will write each of the terms inside the parenthesis separate in form of fraction.

Complete step by step solution:
Step: 1 the given complex number is,
46ii\dfrac{{4 - 6i}}{i}
Multiply the denominator and numerator of the complex number with i to simplify the given complex number.
i×(46i)i×i\Rightarrow \dfrac{{i \times \left( {4 - 6i} \right)}}{{i \times i}}
Multiply the i inside the bracket of the given complex number.
i×(46i)i×i=(4i6i2)i2\Rightarrow \dfrac{{i \times \left( {4 - 6i} \right)}}{{i \times i}} = \dfrac{{\left( {4i - 6{i^2}} \right)}}{{{i^2}}}
We know that the value of i2=1{i^2} = - 1 in the complex number, so substitute i2=1{i^2} = - 1 in the given expression.
(4i6i2)i2=4i6(1)(1)\Rightarrow \dfrac{{\left( {4i - 6{i^2}} \right)}}{{{i^2}}} = \dfrac{{4i - 6\left( { - 1} \right)}}{{\left( { - 1} \right)}}
Now simplify the given express to write it into its simplest form.
4i6(1)(1)=4i+61\dfrac{{4i - 6\left( { - 1} \right)}}{{\left( { - 1} \right)}} = \dfrac{{4i + 6}}{{ - 1}}
Multiply by (1)\left( { - 1} \right) to both the numerator and denominator of complex number.
1×(4i+6)1×1\Rightarrow \dfrac{{ - 1 \times \left( {4i + 6} \right)}}{{ - 1 \times - 1}}
Simplify the number to get the final result.
1×(4i+6)1×1=(6+4i)\Rightarrow \dfrac{{ - 1 \times \left( {4i + 6} \right)}}{{ - 1 \times - 1}} = - \left( {6 + 4i} \right)

Final Answer:
Therefore the simplest form of the given complex number is equal to (6+4i) - \left( {6 + 4i} \right).

Note:
Students are advised to remember the properties of complex numbers. They must use i2=1{i^2} = - 1 while solving the numbers. They should simplify the numbers with help of basic arithmetic operations as they do in normal. They must know that the simplest form of a given complex number is a+iba + ib, where aa is the real part and (ib)\left( {ib} \right) is the imaginary part of the complex number.