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 · 2021-11-09T11:20:52-05:00

The simplified expression is: \(\mathbf{\frac{i+13}{170}}\)

The fraction is given as:

\(\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i}}\)

Collect like terms

\(\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{\sqrt{-1}}{3 -2+8i + 5i}}\)

\(\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{\sqrt{-1}}{1+13i}}\)

In complex numbers, we have:

\(\mathbf{i =\sqrt{-1}}\)

So, we have:

\(\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{i}{1+13i}}\)

Rationalize

\(\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{i}{1+13i} \times \frac{1-13i}{1-13i}}\)

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

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

\(\mathbf{\frac{\sqrt{-1}}{3 +8i -2 + 5i} = \frac{i+13}{170}}\)

Hence, the simplified expression is: \(\mathbf{\frac{i+13}{170}}\)