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Question: How do you simplify \(\dfrac{{3 + i}}{{3 - i}}\) and write the complex number in standard form?...

How do you simplify 3+i3−i\dfrac{{3 + i}}{{3 - i}} and write the complex number in standard form?

Explanation

Solution

Complex number is combination of real part and imaginary part. Complex numbers are written as a+iba + ib where aa is real part and bb is imaginary part. Here is 3+i3 + i , 3 is the real part and 1⋅i1 \cdot i is the imaginary part. To solve this question multiply the given by conjugate.
A conjugate of a complex number is the same number but with an opposite sign. For example, the conjugate of a+iba + ib is a−iba - ib.

Complete step by step answer:
In this question, we multiply both the numerator and denominator by the conjugate of the denominator.
Here, the denominator is 3−i3 - i, Conjugate of denominator 3−i3 - i is 3+i3 + i
Multiplying the numerator and denominator by conjugate 3+i3 + i
⇒3+i3−i×3+i3+i\Rightarrow \dfrac{{3 + i}}{{3 - i}} \times \dfrac{{3 + i}}{{3 + i}}
Solving the above,
⇒(3+i)2(3−i)(3+i)\Rightarrow \dfrac{{{{(3 + i)}^2}}}{{(3 - i)(3 + i)}}
Using(a+b)2=a2+b2+2ab{(a + b)^2} = {a^2} + {b^2} + 2ab and (a−b)(a+b)=a2−b2(a - b)(a + b) = {a^2} - {b^2}
Take 3 as aa and b as ii
⇒(3)2+(i)2+2⋅3⋅i(3)2−(i)2\Rightarrow \dfrac{{{{(3)}^2} + {{(i)}^2} + 2 \cdot 3 \cdot i}}{{{{(3)}^2} - {{(i)}^2}}}
We know, i2=−1{i^2} = - 1
⇒9+(−1)+6i9−(−1)\Rightarrow \dfrac{{9 + ( - 1) + 6i}}{{9 - ( - 1)}}
Add and subtract like terms in both numerator and denominator
⇒8+6i10\Rightarrow \dfrac{{8 + 6i}}{{10}}
It can be written as 810+6i10\dfrac{8}{{10}} + \dfrac{{6i}}{{10}} to make it in the form of a+iba + ib
It can be further simplified as
⇒45+3i5\Rightarrow \dfrac{4}{5} + \dfrac{{3i}}{5}

Thus, simplified 3+i3−i\dfrac{{3 + i}}{{3 - i}} as 45+3i5\dfrac{4}{5} + \dfrac{{3i}}{5} in standard form.

Addition information:
Sometimes complex numbers are represented as ZZ , Re⁡()\operatorname{Re} () for real part and Im⁡()\operatorname{Im} () for imaginary parts.
Like Z=a+ibZ = a + ib , hereRe(z)=aRe\left( z \right) = a and Im⁡(z)=b\operatorname{Im} (z) = b .

Note: Remember the values i2=−1{i^2} = - 1 , i=−1i = \sqrt { - 1} and i3=−i{i^3} = - i .
Split the complex-valued answer into two separate terms that are a real part and an imaginary part to make it easier.
Always write the answer in a standard form that is a+iba + ib.