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Question

Answer

$\frac{( 2 + 3 i ) \cdot ( 1 - 2 i )}{1 - 4 i} + \frac{4 + i}{2 + 3 i} = \frac{233 i + 343}{221}$

Explanation

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$\frac{( 2 + 3 i ) \cdot ( 1 - 2 i )}{1 - 4 i} + \frac{4 + i}{2 + 3 i} \overset{◯}{1} \frac{( 2 + 3 i ) \cdot ( 1 - 2 i )}{1 - 4 i} + \frac{11 - 10 i}{13} \overset{◯}{2} \frac{2 - 4 i + 3 i - 6 i ^{2}}{1 - 4 i} + \frac{11 - 10 i}{13} \overset{◯}{3} \frac{- 6 i ^{2} - i + 2}{1 - 4 i} + \frac{11 - 10 i}{13} \overset{◯}{4} \frac{6 - i + 2}{1 - 4 i} + \frac{11 - 10 i}{13} \overset{◯}{5} \frac{- i + 8}{1 - 4 i} + \frac{11 - 10 i}{13} \overset{◯}{6} \frac{12 + 31 i}{17} + \frac{11 - 10 i}{13} \overset{◯}{7} \frac{233 i + 343}{221}$
①	Divide $4 + i$ by $2 + 3 i$ to get $\frac{11 - 10 i}{13}$ . ( view steps )
②	Multiply each term of $(2 + 3 i)$ by each term in $(1 - 2 i)$ . $(2 + 3 i) \cdot (1 - 2 i) = 2 - 4 i + 3 i - 6 i^{2}$
③	Combine like terms: $2 - 4 i + 3 i - 6 i^{2} = - 6 i^{2} - i + 2$
④	$- 6 i^{2} = - 6 \cdot (- 1) = 6$
⑤	Combine like terms: $6 - i + 2 = - i + 8$
⑥	Divide $8 - i$ by $1 - 4 i$ to get $\frac{12 + 31 i}{17}$ . ( view steps )
⑦	Add $\frac{12 + 31 i}{17}$ and $\frac{11 - 10 i}{13}$ to get $\frac{233 i + 343}{221}$ . To add raitonal expressions, both fractions must have the same denominator. We can create a common denominator by multiplying the first fraction by $13$ and the second by $17$ . $\frac{12 + 31 i}{17} + \frac{11 - 10 i}{13} = \frac{( 12 + 31 i ) \cdot 13}{17 \cdot 13} + \frac{( 11 - 10 i ) \cdot 17}{13 \cdot 17} = = \frac{156 + 403 i}{221} + \frac{187 - 170 i}{221} = \frac{233 i + 343}{221}$

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