Question

1. (-1 - 4i)(11 + 6i)
2. (2 - 3i)²
3. (-5 - 2i)/(6i)
4. 7i/(6 + 2i)

Explanation:

Step1: Expand using distributive property

$(-1-4i)(11+6i) = -1(11) -1(6i) -4i(11) -4i(6i)$

Step2: Calculate each term

$= -11 -6i -44i -24i^2$

Step3: Substitute $i^2=-1$

$= -11 -50i -24(-1)$

Step4: Simplify real terms

$= -11 +24 -50i$
$= 13 -50i$

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Step1: Use square of binomial formula

$(2-3i)^2 = 2^2 - 2(2)(3i) + (3i)^2$

Step2: Calculate each term

$= 4 -12i +9i^2$

Step3: Substitute $i^2=-1$

$= 4 -12i +9(-1)$

Step4: Simplify real terms

$= 4 -9 -12i$
$= -5 -12i$

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Step1: Multiply by conjugate of denominator

$\frac{-5-2i}{6i} \times \frac{-6i}{-6i} = \frac{(-5-2i)(-6i)}{(6i)(-6i)}$

Step2: Expand numerator and denominator

$= \frac{30i +12i^2}{-36i^2}$

Step3: Substitute $i^2=-1$

$= \frac{30i +12(-1)}{-36(-1)}$

Step4: Simplify numerator and denominator

$= \frac{-12 +30i}{36}$

Step5: Split and reduce fractions

$= -\frac{12}{36} + \frac{30}{36}i = -\frac{1}{3} + \frac{5}{6}i$

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Step1: Multiply by conjugate of denominator

$\frac{7i}{6+2i} \times \frac{6-2i}{6-2i} = \frac{7i(6-2i)}{(6+2i)(6-2i)}$

Step2: Expand numerator and denominator

$= \frac{42i -14i^2}{6^2 - (2i)^2}$

Step3: Substitute $i^2=-1$

$= \frac{42i -14(-1)}{36 -4(-1)}$

Step4: Simplify numerator and denominator

$= \frac{14 +42i}{40}$

Step5: Split and reduce fractions

$= \frac{14}{40} + \frac{42}{40}i = \frac{7}{20} + \frac{21}{20}i$

Answer:

1. $13 - 50i$
2. $-5 - 12i$
3. $-\frac{1}{3} + \frac{5}{6}i$
4. $\frac{7}{20} + \frac{21}{20}i$