Simplify the expression fraction with numerator of the square root of negative one and denominator of the quantity three plus eight times i minus the quantity two plus five times i.

fraction with numerator x-3 plus i and denominator 10
fraction with numerator negative 3 plus i and denominator 10
fraction with numerator 3 minus i and denominator 10
fraction with numerator negative 3 minus i and denominator 10

The right answer for the question that is being asked and shown above is that: "fraction with numerator negative 3 minus i and denominator 10." This is the simplified expression fraction with numerator of the square root of negative one and denominator of the quantity three plus eight times i minus the quantity two plus five times i.

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MrRoyal MrRoyal · Answer 2 · 2021-11-09T17:20:52+01:00

The simplified expression is: [tex]\mathbf{\frac{i+13}{170}}[/tex]

The fraction is given as:

[tex]\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i}}[/tex]

Collect like terms

[tex]\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{\sqrt{-1}}{3 -2+8i + 5i}}[/tex]

[tex]\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{\sqrt{-1}}{1+13i}}[/tex]

In complex numbers, we have:

[tex]\mathbf{i =\sqrt{-1}}[/tex]

So, we have:

[tex]\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{i}{1+13i}}[/tex]

Rationalize

[tex]\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{i}{1+13i} \times \frac{1-13i}{1-13i}}[/tex]

[tex]\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{i(1-13i)}{1^2-(13i)^2}}[/tex]

[tex]\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{i-13(-1)}{1-169(-1)}}[/tex]

[tex]\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{i+13}{170}}[/tex]

Hence, the simplified expression is: [tex]\mathbf{\frac{i+13}{170}}[/tex]