Question

1. - / 2 pointsdetailswrite the complex conjugate of the complex number.$sqrt{-13}$13multiply the number by its complex conjugate. (simplify your answer completely.)tutorialresourcesebookviewing saved work revert to last responsesubmit answer6. - / 2 pointsdetailswrite the complex conjugate of the complex number.$1 + sqrt{8}$multiply the number by its complex conjugate. (simplify your answer completely.)resourcesebooksubmit answer7. - / 1 pointsdetailswrite the quotient in standard form.$\frac{9 - 17i}{1 - 3i}$resourcesebooksubmit answer8. - / 1 pointsdetailsperform the operation and write the result in standard form.$\frac{2i}{2 + i} + \frac{5}{2 - i}$resourcesebook

Explanation:

Problem 5

Step1: Rewrite number in standard form

$\sqrt{-13} = i\sqrt{13}$

Step2: Find complex conjugate

Complex conjugate of $i\sqrt{13}$ is $-i\sqrt{13}$

Step3: Multiply number by its conjugate

$i\sqrt{13} \times (-i\sqrt{13}) = -i^2 (\sqrt{13})^2$
Since $i^2=-1$, substitute:
$-(-1)(13) = 13$

Problem 6

Step1: Identify complex number form

$1+\sqrt{8}$ is $1+\sqrt{8}+0i$

Step2: Find complex conjugate

Complex conjugate of $1+\sqrt{8}$ is $1-\sqrt{8}$

Step3: Multiply number by its conjugate

Use $(a+b)(a-b)=a^2-b^2$:
$(1+\sqrt{8})(1-\sqrt{8}) = 1^2 - (\sqrt{8})^2 = 1 - 8 = -7$

Problem 7

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $1+3i$:
$\frac{(9-17i)(1+3i)}{(1-3i)(1+3i)}$

Step2: Expand numerator

$9(1) + 9(3i) -17i(1) -17i(3i) = 9 +27i -17i -51i^2$
Simplify: $9 +10i -51(-1) = 9+10i+51=60+10i$

Step3: Expand denominator

$1^2 - (3i)^2 = 1 -9i^2 =1 -9(-1)=10$

Step4: Simplify the fraction

$\frac{60+10i}{10} = 6 + i$

Problem 8

Step1: Simplify each fraction separately

First fraction: $\frac{2i}{2+i} \times \frac{2-i}{2-i} = \frac{2i(2-i)}{(2)^2-(i)^2} = \frac{4i-2i^2}{4-(-1)} = \frac{4i+2}{5}$
Second fraction: $\frac{5}{2-i} \times \frac{2+i}{2+i} = \frac{5(2+i)}{(2)^2-(i)^2} = \frac{10+5i}{5}=2+i$

Step2: Add the two results

$\frac{4i+2}{5} + 2+i = \frac{2}{5}+2 + \frac{4i}{5}+i = \frac{2}{5}+\frac{10}{5} + \frac{4i}{5}+\frac{5i}{5} = \frac{12}{5} + \frac{9i}{5}$

Answer:

1. Problem 5:

• Complex conjugate: $-i\sqrt{13}$
• Product: $13$

1. Problem 6:

• Complex conjugate: $1-\sqrt{8}$
• Product: $-7$

1. Problem 7: $6 + i$
2. Problem 8: $\frac{12}{5} + \frac{9}{5}i$