Question
Question: How do you simplify \(\dfrac{{4 - 6i}}{i}\)?...
How do you simplify i4−6i?
Solution
The question belongs to the simplification of complex numbers. Any fraction of complex numbers can be represented in only (a+ib) form by doing some basic arithmetic operations. We know that the value of $${i^2} = - 1inthecomplexnumbers.Wecanreplaceminussignsbysimplyreplacingitwith i2 $. 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,
i4−6i
Multiply the denominator and numerator of the complex number withi to simplify the given complex number.
⇒i×ii×(4−6i)
Multiply the i inside the bracket of the given complex number.
⇒i×ii×(4−6i)=i2(4i−6i2)
We know that the value of i2=−1 in the complex number, so substitute i2=−1 in the given expression.
⇒i2(4i−6i2)=(−1)4i−6(−1)
Now simplify the given express to write it into its simplest form.
(−1)4i−6(−1)=−14i+6
Multiply by (−1) to both the numerator and denominator of complex number.
⇒−1×−1−1×(4i+6)
Simplify the number to get the final result.
⇒−1×−1−1×(4i+6)=−(6+4i)
Final Answer:
Therefore the simplest form of the given complex number is equal to −(6+4i).
Note:
Students are advised to remember the properties of complex numbers. They must use i2=−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+ib, where a is the real part and (ib) is the imaginary part of the complex number.