Complex Numbers

Problem 201

Find the difference of the complex numbers in the complex plane. $-5 i-(6-8 i)$ $\square$

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Problem 202

Situation problème 2 (à résoudre avec les élèves) Mensieur KOFF* ’ossède 3 terrains dont il veut abso'...ment clôturer car il lui est rapporté que des individus mal intent':onnés utilisent zes espaces non occupés à de mârvaises fins. Гee dernier décide donc d’acheter du fil barbelé pour clôturer ces 3 terrains. ie rculeau de 5 m de fil est vend:ı à 3500 f. -le $1^{\text {or }}$ terrain est formé de !'ensemble des points $M(z)$ tel que $|2 i z-1-3 i|=8$ -le $2^{\text {èmc }}$ terrain quant à lui a une forme rectangulaire dont les dimensions sont la partie réelle et la partie imaginaire solution de l'équation : $(1+4 i) z+(3-4 i) \bar{z}=4-3 i$ ou $\bar{z}$ est le conjugué de $z$ -le $3^{c}$ terrain est formé de l'ensemble des points $M(z)$ culi plan complexe tel que $\operatorname{Re}\left(z^{\prime}\right)=0$ avec $Z=\frac{z}{z+4 i}$ Les distances sont tous en décamètre. A partir de tes connaissances détermines le montant total à dépenser par monsieur Koffi pour la c'otire de ses terrains

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Problem 203

Find the product, $A B$ , of the two complex numbers.

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Problem 204

Find the product. $(-5-2 i)(-5+2 i)$ $(-5-2 i)(-5+2 i)=$ $\square$ (Type your answer in the form a $+b i$ .)

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Problem 205

Find the sum of the pair of complex numbers. Then graph both complex numbers and their resultant. $5-6 i,-2+4 i$
The sum is $\square$ (Type your answer in the form $\mathrm{a}+\mathrm{bi}$ .)

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Problem 206

Write the complex number in trigonometric form $\mathrm{r}(\boldsymbol{\operatorname { c o s }} \theta+i \boldsymbol{\operatorname { s i n }} \theta)$ , with $\theta$ in the interval $\left[0^{\circ}, 360^{\circ}\right)$ . $-3-3 i$ $-3-3 i=\square\left(\cos \square^{\circ}+i \boldsymbol{\operatorname { s i n }} \square^{\circ}\right)$ (Type the value for $r$ as an exact answer, using radicals as needed. Type the value for $\theta$ as an integer or decimal rounded to the nearest tenth as needed.)

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Problem 207

Solve for $x$ and $y$ in the system: $9+(x+y)i=9-i$ and $x+6i=9+6i$ .

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Problem 208

Calculate and simplify: $(1+\sqrt{-49})-(29 i-34)$ in the form $a+bi$ .

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Problem 209

Find the square root of the number. $\sqrt{-24}$
The square root is $\square$

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Problem 210

10. What is the imaginary part of the complex number $7-5 i$ ? a. 7 b. -5 c. 5 d. 0 a b C d not given

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Problem 211

20. Which complex number lies on the imaginary axis? a. $4+0 i$ b. $0+3 i$ c. $-2+2 i$ d. $1-i$ a b C d not given

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Problem 212

8. The value of $\frac{(1-i)^{8}(\sqrt{3}-i)^{3}}{(1+i)^{14}}$ is: a. -1 b. 1 c. $i$ d. $-i$ e. $i-1$

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Problem 213

Simplify $-5-\sqrt{-44}$ . $-5-4 \sqrt{11 i}$ $-5-4 i \sqrt{11}$ $-5-2 i \sqrt{11}$ $-5-2 \sqrt{11 i}$

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Problem 214

Simplify $\sqrt{-100}$ . $-10$ $-10 i$ $10 i$ 10

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Problem 215

Simplify the expression to a + bi form: $-2 i^{19}-10 i^{116}-10 i^{17}+i^{85}$

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Problem 216

Simplify the expression to a + bi form: $-10 i^{118}+5 i^{43}+4 i^{16}+3 i^{86}$

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Problem 217

Find the product of the complex numbers $2+7i$ and $3-2i$ . What is the result?

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Problem 218

Find the product and express it as $a + b i$ : $(7 - 2 i)(1 + i)$ .

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Problem 219

Evaluate the quotient and express it as $a + b i$ : $\frac{4 - 3 i}{1 - 4 i}$

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Problem 220

Evaluate the quotient and express it as $a$ : $\frac{3-7 i}{1-3 i}$

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Problem 221

17) $(9 i)-(-6 i+10)=$ 18) $(12 i+8)+(-7 i)=$ 19) $(13 i)-(17+3 i)=$ 20) $(3+5 i)+(8+3 i)=$ 21) $(8-3 i)+(4+i)=$ 22) $(10+9 i)+(6+8 i)=$

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Problem 222

Which of the following is equivalent to $i^{26}$ ? -1 $-i$ 1 i

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Problem 223

Perform the operation. Write the answer in standard form. $(4+i)(2-i)$ $\square$

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Problem 224

Question Watch Video Show Ex
Simplify the expression to a + bi form: $(2+i)+(-11+2 i)$
Answer Attempt 1 out of 2 $\square$ Submit An

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Problem 225

Question Watch Vi
Simplify the expression to a + bi form: $(-4-8 i)+(-9-8 i)$

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Problem 226

Find the sum of the pair of complex numbers. $\frac{1}{2}+\frac{5}{6} i, \frac{5}{6}+\frac{1}{2} i$
The sum is $\square$ . (Simplify your answer. Type your answer in the form a + bi. Use integers or fractions for any numbers in the expressic

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Problem 227

Write the quotient in rectangular form. $\frac{4 \text { cis } 400^{\circ}}{20 \text { cis } 85^{\circ}}$ $\frac{4 \text { cis } 400^{\circ}}{20 \text { cis } 85^{\circ}}=$ $\square$ (Simplify your answer, including any radicals. Use integers or fr

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Problem 228

Given $z=6+2 i$ , prove that $z-i z=8-4 i$ and $|z|^{2}+|i z|^{2}=|z-i z|^{2}$ .

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Problem 229

Find the square root of $-5+12 i$ in the form $a+bi$ , where $a, b \in \mathbb{R}$ .

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Problem 230

Find the product and write the result in standard form. $(3-8 i)(-7-2 i)$

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Problem 231

5 A B C D F G H I $2 e^{\left(90^{\circ}\right) \mathrm{i}}$ A) SELECT ALI APPLICABIE CHO B) $2\left(\cos \left(90^{\circ}\right)+i \sin \left(90^{\circ}\right)\right) \quad 2 \cos \left(90^{\circ}\right)+i \cdot 2 \sin \left(90^{\circ}\right)$ C) D) $-\cos \left(90^{\circ}\right)-i \sin \left(90^{\circ}\right) \quad 2\left(\cos \left(90^{\circ}\right)-i \sin \left(90^{\circ}\right)\right)$ E) $\approx 1+i(2)$

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Problem 232

$12 \cos \left(330^{\circ}\right)+i \cdot 12 \sin \left(330^{\circ}\right)$ A) $12 e^{\left(510^{\circ}\right) i}$ C) B) $12 e^{(330 \%) i}$ D)

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Problem 233

10 A B C D E F G I $\sqrt[3]{10 e^{i\left(420^{3}\right)}}$ A) SEIECT ALI APPICA $10^{\frac{1}{2}} e^{i\left(\frac{\infty}{3}\right)}$ B) $\sqrt[3]{10}\left[\cos \left(420^{\circ}\right)+i \sin \left(420^{\circ}\right)\right]$ C) D) $e^{i\left(\frac{a P}{2}\right)}$ $\sqrt[3]{10} e^{i\left(120^{\circ}\right)}$ E) $\sqrt[3]{10}\left[\cos \left(\frac{420^{\circ}}{3}\right)+i \sin \left(\frac{420^{\circ}}{3}\right)\right]$ F) $10\left[\cos \left(\frac{420^{\circ}}{3}\right)+i \sin \left(\frac{420^{\circ}}{3}\right)\right]$

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Problem 234

3) $\sqrt{-150}$

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Problem 235

Simplify. $\sqrt{-2}$

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Problem 236

Write the expression as a complex number in standard form: $(-2+5 i)(-1+4 i)$ $15-18 i$ $-18-13 i$ $-13-18 i$ $-18+13 i$

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Problem 237

15. Divide the complex numbers. Write the answer in the form $a+b i$ . $\frac{1-9 i}{2-8 i}$

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Problem 238

Convert the complex number to polar form: $4-4 i$

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Problem 239

Subtract. $(-8 j-6)-(-9 j+6)$

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Problem 240

Balanced the following skeleton ionic reaction in basic solution: $\mathrm{MnO}_{4}^{-}(a q)+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}(a q) \rightarrow \mathrm{MnO}_{4}^{2-}(a q)+\mathrm{CO}_{3}^{2-}(a q)$
Hint: sample problem 21.1. It is easier to balance the reaction on a piece of paperand then fill in the following table. A, B, D and E are coefficients of the reactants/products; In order to balance the reaction some substances may need to be added, as $\mathbf{C}$ and $\mathrm{F}(\mathrm{H}+, \mathrm{OH}-\mathrm{H} 2 \mathrm{O}$ , wtih coeifficents). If nothing to add, choose NA (1) Divide the reaction into two half-reactions (2) Balance the atoms and charges in each half-reaction. Oxygen can be balanced by adding $\mathrm{H}_{2} \mathrm{O}$ to the other side, and $\mathrm{H}^{+}$ needs to be added to the side with Oxygen; charges can be balanced by adding or subtracting the number of electrons matching the oxidation number change. (3) multiply each halfreaction by an integer so that the number of e lost equals the number of e gained. (4) add the half-reactions. (5)Add OHto both sides of the equation to neutralize $\mathrm{H}+$ , and cancel H 2 O . \begin{tabular}{|l|l|l|l|l|l|} \hline \multicolumn{4}{|c|}{ Reactants (with coefficients) } & \multicolumn{2}{c|}{ Products (with coefficients) } \\ \hline $\left[\right.$ A] $\mathrm{MnO}_{4}^{-}$ & [B] $\mathrm{C}_{2} \mathrm{O}_{4}^{2-}$ & [C] & [D] $\mathrm{MnO}_{4}^{2-}$ & [E] $\mathrm{CO}_{3}^{2-}$ & [F] \\ \hline \end{tabular}

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Problem 241

Create a quadratic equation with real coefficients that has complex solutions. Explain your reasoning. (5.00 Points)

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Problem 242

When the following equation is balanced properly under basic conditions, what are the coefficients of the species shown? $\square$ $\square$ $\mathrm{SO}_{3}{ }^{2-} \longrightarrow$ $\square$ $\square$ $F_{2}$
Water appears in the balanced equation as a $\square$ (reactant, product, neither) with a coefficient of $\square$ (Enter 0 for neither.)
How many electrons are transferred in this reaction? $\square$

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Problem 243

9. Simplify: $\sqrt{-50}=$ $\qquad$

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Problem 244

$7i$ $\boxed{\phantom{7}}(\cos(\boxed{\phantom{90^\circ}})^\circ) + i \sin(\boxed{\phantom{90^\circ}})^\circ))$

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Problem 245

că $(\boldsymbol{Z}, \mathrm{o},-)$ este un domeniu de integritate și să se afle $U(\boldsymbol{Z})$ ;
2. Fie $\boldsymbol{Z}[i \sqrt{2}]=\{z / z=x+j y \sqrt{2} ; x, y \in \boldsymbol{Z}\}$ și $\delta: Z[i \sqrt{2}] \rightarrow N, \delta(x+j y \sqrt{2})=x^{2}+2 y^{2}$ ; $\forall x, y \in \boldsymbol{Z}$ . a) Sai se afle elementele asociate lui $z \in \boldsymbol{Z}[i \sqrt{2}]$ ; b) Să se arate cá 5 este ireductibil, iar 3 este reductibil in $\boldsymbol{Z} i \sqrt{2}]$ .

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Problem 246

MARCH 2016
11. a) Which one of the following is the real part and imaginary parts of the complex number: $\left(\frac{1+i}{1-i}\right)-\left(\frac{1-i}{1+i}\right)$ ? i) 0 and 1 ii) 0 and 2 iii) 3 and 2 iv) 0 and 4 (1) b) Deleted. c) Deleted.
IMPROVEMENT 2015
12. a) What is $i^{-35}$ ? b) Deleted. c) Deleted.
MARCH 2015 Deleted. IMPROVEMENT 2014 Deleted. MARCH 2014 Deleted. SEPTEMBER 2013
13. a) Express $\frac{1+i}{1-i}$ in the form $a+i b$ . b) Deleted.
MARCH 2013 Deleted. (2)
MARCH 2011
22. Consider the complex number $Z=\frac{2+i}{(1+i)(1-2 i)}$ a) Express Z in the form $a+i b$ . b) Deleted.
MARCH 2010
23. i) Express the complex number $z=\frac{5+i}{2+3 i}$ in the form $a+i b$ ii) Deleted.
AUGUST 2009
24. i) Express the complex number $\frac{3-\sqrt{-16}}{1-\sqrt{-9}}$ in the form $a+i b$ . ii) Deleted. iii) Deleted.
MARCH 2009
25. a) Express the complex number $\frac{2-i}{(1-i)(1+2 i)}$ in the form $a+i b$ . b) Deleted. c) Deleted.

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Problem 247

Rewrite $\frac{8-i}{3-2 i}$ as $a+bi$ and find the value of $a$ (with $i=\sqrt{-1}$ ).

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Problem 248

Rewrite the expression $\frac{8-i}{3-2 i}$ as $a+bi$ . What is the value of $a$ ? A) 2 B) $\frac{8}{3}$ C) 3 D) $\frac{11}{3}$

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Problem 249

Number Systems Complex Numbers
11. Evaluate $1 + \frac{1+i}{\sqrt{2}} + \left(\frac{1+i}{\sqrt{2}}\right)^2 + \left(\frac{1+i}{\sqrt{2}}\right)^3 + \left(\frac{1+i}{\sqrt{2}}\right)^4$ .

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Problem 250

54. Seja $z = \frac{i}{i+1} + \frac{1}{i}$
a) Obtenha a forma algébrica e a trigonométrica de $z$ .
b) Qual é a forma trigonométrica de $z^2$ ?

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Problem 251

56. Obtenha a forma algébrica de cada um dos seguintes números complexos: a) $z = 4(\cos 120^\circ + i \sin 120^\circ)$ b) $z = 3(\cos 90^\circ + i \sin 90^\circ)$ c) $z = \cos 210^\circ + i \sin 210^\circ$ d) $z = \sqrt{2}(\cos 135^\circ + i \sin 135^\circ)$

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Problem 252

Perform the indicated operation 14.) $(1+5 i)+(6-i)$

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Problem 253

Find the product and express it as $a + b i$ : simplify $(2 + 5 i)^{2}$ .

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Problem 254

Find the difference and express it as $a + b i$ : $(-3 + 2 i) - (4 - 3 i)$

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Problem 255

Evaluate $(1-2 i)(4-1 i)$ and express the result as $a + b i$ .

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Problem 256

Simplify $i^{49}$ . What is the result? Options: $1$ , $i$ , $-i$ , $-1$ .

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Problem 257

Divide $\frac{-3+1 i}{2 i}$ and express the result as $a + b i$ .

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Problem 258

(b) Simplify each of the following: (i) $\sqrt{-28}$ (ii) $\sqrt{-108}$ (iii) $\sqrt{-180}$ (iv) $\sqrt{-44}$ (v) $\sqrt{-200}$ (vi) $\sqrt{-1445}$ (vii) $\sqrt{-48}$ (viii) $\sqrt{-50}$ (ix) $\sqrt{-52}$ (x) $\sqrt{-675}$ (xi) $\sqrt{-2023}$

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Problem 259

Final Question 2 of 45
Multiply. $(6+3 i)(6+8 i)$

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Problem 260

Think about plotting points in the complex plane to represent the following numbers: $-3+8 i$ $4 i$ 6 5-2i
Where is each point located on the graph? $-3+8 \mathrm{i}$ is $\square$ $4 i$ is $\square$ 6 is $\square$ $5-2 i$ is $\square$ DONE

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Problem 261

$(12-9 i)+(32-6 i)=$ $\square$ Express your answer in the form $(a+b i)$ .

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Problem 262

Evaluate the expression $(-6+7 i)+(-8-1 i)$ and write the result in the form $a+b i$ . The real number $a$ equals $\square$ The real number $b$ equals $\square$

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Problem 263

For numbers 3-6, simplify.
3. $\sqrt{-32}$ $32 / i$

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Problem 264

What is the sum of the complex numbers $\sqrt{-9}$ and $\sqrt{-16}$ ? Choose from: A. $7 i$ , B. $5 i$ , C. 7, D. 5, E. -5.

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Problem 265

2. $(7 + 3i) - (5 - 6i)$

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Problem 266

10 Use the properties of complex numbers to simplify $(9+\sqrt{-4})+(-9-\sqrt{-16})$ . (A) $-2 i$ (B) $18-\sqrt{-20}$ (C) $18-2 i$ (D) $6 i$

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Problem 267

9 Solve the equation $-(x-3)^{2}-3=7$ and write the answer as a complex number in the standard form $a \pm b i$ . (A) -7 (B) $-3 \pm i \sqrt{10}$ (C) $3 \pm i \sqrt{10}$ (D) $3 \pm 10 i$

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Problem 268

Question 7 of 10 A number written in the form $a + bi$ is called a _______ number.
A. complex B. quadratic C. linear D. discriminant SUBMIT

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Problem 269

$(4+i)(2-5 i)$

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Problem 270

32. Write $5(\cos 146^\circ + i\sin 146^\circ)$ in standard $a + bi$ form. Round to two decimal places.
A. $2.80 + 4.15i$ B. $-4.15 - 2.80i$ C. $-4.15 + 2.80i$ D. $-1.35 - 1.35i$

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Problem 271

6. $(6+5i)+(6-2i)$

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Problem 272

4. $(2 + 3i) - (5 - 2i)$ $i$ $3i$ 5 10 $15i$ $-10i$ $-9i$ $-6i^2$

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Problem 273

b. $(-10+i)(3-9 i)$ c. $(3-4 i)-(-5-2 i)$

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Problem 274

Find the value of $i^{99}$ .

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Problem 275

Question 12 ot 25 16 Final Exam
Find the following product, and write the product in rectangular form. $\left(\sqrt{2} \text { cis } 30^{\circ}\right)\left(\sqrt{2} \text { cis } 60^{\circ}\right)$ $\left(\sqrt{2}\right.$ 6ie $\left.30^{\circ}\right)\left(\sqrt{2}+60^{\circ}\right)=$ $\square$

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Problem 276

1. $\text { 1. } \sqrt{-25} \cdot \sqrt{-15}$

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Problem 277

Find the value of $i^{120}$ .

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Problem 278

5. $\frac{5 i}{-5+7 i}$ simplify

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Problem 279

Page 17 of 35 Previous Next
Halla las raíces cúbicas del siguiente número - 729.

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Problem 280

Add: $(8-\sqrt{-16})+(-20+\sqrt{-49})$ . Find the sum.

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Problem 281

Calculate the difference: $(8-4 i)-(-20+7 i)$ , find the result.

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Problem 282

Calculate the product of the complex numbers: $(-8+10 i)(-5+2 i)$ .

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Problem 283

Evaluate $\frac{-2-2 i}{4 i}$ and express the result as $a + b i$ .

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Problem 284

Simplify the expression $\frac{1}{1+5 i}+\frac{2}{-3 i}$ . Enter the answer as a reduced fraction.

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Problem 285

$2+\sqrt{-98}$

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Problem 286

Simplify the complex number $i^{0.52}$ .

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Problem 287

Simplify the expression: $\frac{-20+\sqrt{-75}}{40}$ .

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Problem 288

Simplify and express in standard form: $\frac{-20+\sqrt{-75}}{40}$ .

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Problem 289

14. Given the following complex numbers find $z_{1}+z_{2}, z_{1} \cdot z_{2}$ and $\frac{z_{1}}{z_{2}}$ : $\begin{array}{l} z_{1}=-3-4 i \\ z_{2}=-2 \sqrt{3}+2 i \end{array}$

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Problem 290

Solve the equation $i^{2}+8 i+16=0$ .

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Problem 291

9. What is $(2 i-5)(4+7 i)$ in simplest form? $14 i^{2}-27 i-20$ $57 i+20$ $-27 i-34$ $-27 i-6$ Clear All

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Problem 292

Simplify. $-\left(-3+\frac{i}{5}\right)+\frac{i}{15}-\frac{3+3 i}{5}$
Write your answer in the form a + bi. Simplify all fractions. $\square$

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Problem 293

Write $(-5+10 i)^{2}$ in simplest $a+b i$ form.

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Problem 294

23. What is $\sqrt{-144}$ ?

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Problem 295

$(4-i)(2+5 i)$ a. $2(4+9 i)$ c. $3+18 i$ b. $(13+18 i)$ d. $(8+18 i)$

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Problem 296

my smplity. $(6 \sqrt{-4})(\sqrt{-1})$

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Problem 297

13. Simplify $(5+2 i)(1+3 i)$ . A. $-1+17 i$ B. -1 C. $5+6 i$ D. $11+17 i$

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Problem 298

18. Perform the operations of complex numbers and simplify: $(1-4 i)(5-2 i)-(2-i)$

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Problem 299

$\frac{4-3 i}{-5-3 i}$

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Problem 300

32. Multiply. Write the answer in standard form. $15 i(-1+2 i)$

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1 2 3

Complex Numbers

Problem 201

Find the difference of the complex numbers in the complex plane. −5i−(6−8i)-5 i-(6-8 i)−5i−(6−8i) □\square□

Problem 202

Problem 203

Find the product, ABA BAB, of the two complex numbers.

Problem 204

Find the product. (−5−2i)(−5+2i)(-5-2 i)(-5+2 i)(−5−2i)(−5+2i) (−5−2i)(−5+2i)=(-5-2 i)(-5+2 i)=(−5−2i)(−5+2i)= □\square□ (Type your answer in the form a +bi+b i+bi.)

Problem 205

Find the sum of the pair of complex numbers. Then graph both complex numbers and their resultant. 5−6i,−2+4i5-6 i,-2+4 i5−6i,−2+4iThe sum is □\square□ (Type your answer in the form a+bi\mathrm{a}+\mathrm{bi}a+bi.)

Problem 206

Problem 207

Solve for xxx and yyy in the system: 9+(x+y)i=9−i9+(x+y)i=9-i9+(x+y)i=9−i and x+6i=9+6ix+6i=9+6ix+6i=9+6i.

Problem 208

Calculate and simplify: (1+−49)−(29i−34)(1+\sqrt{-49})-(29 i-34)(1+−49​)−(29i−34) in the form a+bia+bia+bi.

Problem 209

Find the square root of the number. −24\sqrt{-24}−24​The square root is □\square□

Problem 210

10. What is the imaginary part of the complex number 7−5i7-5 i7−5i ? a. 7 b. -5 c. 5 d. 0 a b C d not given

Problem 211

20. Which complex number lies on the imaginary axis? a. 4+0i4+0 i4+0i b. 0+3i0+3 i0+3i c. −2+2i-2+2 i−2+2i d. 1−i1-i1−i a b C d not given

Problem 212

8. The value of (1−i)8(3−i)3(1+i)14\frac{(1-i)^{8}(\sqrt{3}-i)^{3}}{(1+i)^{14}}(1+i)14(1−i)8(3​−i)3​ is: a. -1 b. 1 c. iii d. −i-i−i e. i−1i-1i−1

Problem 213

Simplify −5−−44-5-\sqrt{-44}−5−−44​. −5−411i-5-4 \sqrt{11 i}−5−411i​ −5−4i11-5-4 i \sqrt{11}−5−4i11​ −5−2i11-5-2 i \sqrt{11}−5−2i11​ −5−211i-5-2 \sqrt{11 i}−5−211i​

Problem 214

Simplify −100\sqrt{-100}−100​. −10-10−10 −10i-10 i−10i 10i10 i10i 10

Problem 215

Simplify the expression to a + bi form: −2i19−10i116−10i17+i85-2 i^{19}-10 i^{116}-10 i^{17}+i^{85}−2i19−10i116−10i17+i85

Problem 216

Simplify the expression to a + bi form: −10i118+5i43+4i16+3i86-10 i^{118}+5 i^{43}+4 i^{16}+3 i^{86}−10i118+5i43+4i16+3i86

Problem 217

Find the product of the complex numbers 2+7i2+7i2+7i and 3−2i3-2i3−2i. What is the result?

Problem 218

Find the product and express it as a+bia + b ia+bi: (7−2i)(1+i)(7 - 2 i)(1 + i)(7−2i)(1+i).

Problem 219

Evaluate the quotient and express it as a+bia + b ia+bi: 4−3i1−4i\frac{4 - 3 i}{1 - 4 i}1−4i4−3i​

Problem 220

Evaluate the quotient and express it as aaa: 3−7i1−3i\frac{3-7 i}{1-3 i}1−3i3−7i​

Problem 221

Problem 222

Which of the following is equivalent to i26i^{26}i26 ? -1 −i-i−i 1 i

Problem 223

Perform the operation. Write the answer in standard form. (4+i)(2−i)(4+i)(2-i)(4+i)(2−i) □\square□

Problem 224

Question Watch Video Show ExSimplify the expression to a + bi form: (2+i)+(−11+2i)(2+i)+(-11+2 i)(2+i)+(−11+2i)Answer Attempt 1 out of 2 □\square□ Submit An

Problem 225

Question Watch ViSimplify the expression to a + bi form: (−4−8i)+(−9−8i)(-4-8 i)+(-9-8 i)(−4−8i)+(−9−8i)

Problem 226

Find the sum of the pair of complex numbers. 12+56i,56+12i\frac{1}{2}+\frac{5}{6} i, \frac{5}{6}+\frac{1}{2} i21​+65​i,65​+21​iThe sum is □\square□ . (Simplify your answer. Type your answer in the form a + bi. Use integers or fractions for any numbers in the expressic

Problem 227

Problem 228

Given z=6+2iz=6+2 iz=6+2i, prove that z−iz=8−4iz-i z=8-4 iz−iz=8−4i and ∣z∣2+∣iz∣2=∣z−iz∣2|z|^{2}+|i z|^{2}=|z-i z|^{2}∣z∣2+∣iz∣2=∣z−iz∣2.

Problem 229

Find the square root of −5+12i-5+12 i−5+12i in the form a+bia+bia+bi, where a,b∈Ra, b \in \mathbb{R}a,b∈R.

Problem 230

Find the product and write the result in standard form. (3−8i)(−7−2i)(3-8 i)(-7-2 i)(3−8i)(−7−2i)

Problem 231

Problem 232

12cos⁡(330∘)+i⋅12sin⁡(330∘)12 \cos \left(330^{\circ}\right)+i \cdot 12 \sin \left(330^{\circ}\right)12cos(330∘)+i⋅12sin(330∘) A) 12e(510∘)i12 e^{\left(510^{\circ}\right) i}12e(510∘)i C) B) 12e(330%)i12 e^{(330 \%) i}12e(330%)i D)

Problem 233

Problem 234

3) −150\sqrt{-150}−150​

Problem 235

Simplify. −2\sqrt{-2}−2​

Problem 236

Write the expression as a complex number in standard form: (−2+5i)(−1+4i)(-2+5 i)(-1+4 i)(−2+5i)(−1+4i) 15−18i15-18 i15−18i −18−13i-18-13 i−18−13i −13−18i-13-18 i−13−18i −18+13i-18+13 i−18+13i

Problem 237

15. Divide the complex numbers. Write the answer in the form a+bia+b ia+bi. 1−9i2−8i\frac{1-9 i}{2-8 i}2−8i1−9i​

Problem 238

Convert the complex number to polar form: 4−4i4-4 i4−4i

Problem 239

Subtract. (−8j−6)−(−9j+6)(-8 j-6)-(-9 j+6)(−8j−6)−(−9j+6)

Problem 240

Problem 241

Create a quadratic equation with real coefficients that has complex solutions. Explain your reasoning. (5.00 Points)

Problem 242

Problem 243

9. Simplify: −50=\sqrt{-50}=−50​= \qquad

Problem 244

Find the difference of the complex numbers in the complex plane. $-5 i-(6-8 i)$ $\square$

Find the product, $A B$ , of the two complex numbers.

Find the product. $(-5-2 i)(-5+2 i)$ $(-5-2 i)(-5+2 i)=$ $\square$ (Type your answer in the form a $+b i$ .)

Find the sum of the pair of complex numbers. Then graph both complex numbers and their resultant. $5-6 i,-2+4 i$
The sum is $\square$ (Type your answer in the form $\mathrm{a}+\mathrm{bi}$ .)

Solve for $x$ and $y$ in the system: $9+(x+y)i=9-i$ and $x+6i=9+6i$ .

Calculate and simplify: $(1+\sqrt{-49})-(29 i-34)$ in the form $a+bi$ .

Find the square root of the number. $\sqrt{-24}$
The square root is $\square$

10. What is the imaginary part of the complex number $7-5 i$ ? a. 7 b. -5 c. 5 d. 0 a b C d not given

20. Which complex number lies on the imaginary axis? a. $4+0 i$ b. $0+3 i$ c. $-2+2 i$ d. $1-i$ a b C d not given

8. The value of $\frac{(1-i)^{8}(\sqrt{3}-i)^{3}}{(1+i)^{14}}$ is: a. -1 b. 1 c. $i$ d. $-i$ e. $i-1$

Simplify $-5-\sqrt{-44}$ . $-5-4 \sqrt{11 i}$ $-5-4 i \sqrt{11}$ $-5-2 i \sqrt{11}$ $-5-2 \sqrt{11 i}$

Simplify $\sqrt{-100}$ . $-10$ $-10 i$ $10 i$ 10

Simplify the expression to a + bi form: $-2 i^{19}-10 i^{116}-10 i^{17}+i^{85}$

Simplify the expression to a + bi form: $-10 i^{118}+5 i^{43}+4 i^{16}+3 i^{86}$

Find the product of the complex numbers $2+7i$ and $3-2i$ . What is the result?

Find the product and express it as $a + b i$ : $(7 - 2 i)(1 + i)$ .

Evaluate the quotient and express it as $a + b i$ : $\frac{4 - 3 i}{1 - 4 i}$

Evaluate the quotient and express it as $a$ : $\frac{3-7 i}{1-3 i}$

Which of the following is equivalent to $i^{26}$ ? -1 $-i$ 1 i

Perform the operation. Write the answer in standard form. $(4+i)(2-i)$ $\square$

Question Watch Video Show Ex
Simplify the expression to a + bi form: $(2+i)+(-11+2 i)$
Answer Attempt 1 out of 2 $\square$ Submit An

Question Watch Vi
Simplify the expression to a + bi form: $(-4-8 i)+(-9-8 i)$

Find the sum of the pair of complex numbers. $\frac{1}{2}+\frac{5}{6} i, \frac{5}{6}+\frac{1}{2} i$
The sum is $\square$ . (Simplify your answer. Type your answer in the form a + bi. Use integers or fractions for any numbers in the expressic

Given $z=6+2 i$ , prove that $z-i z=8-4 i$ and $|z|^{2}+|i z|^{2}=|z-i z|^{2}$ .

Find the square root of $-5+12 i$ in the form $a+bi$ , where $a, b \in \mathbb{R}$ .

Find the product and write the result in standard form. $(3-8 i)(-7-2 i)$

$12 \cos \left(330^{\circ}\right)+i \cdot 12 \sin \left(330^{\circ}\right)$ A) $12 e^{\left(510^{\circ}\right) i}$ C) B) $12 e^{(330 \%) i}$ D)

3) $\sqrt{-150}$

Simplify. $\sqrt{-2}$

Write the expression as a complex number in standard form: $(-2+5 i)(-1+4 i)$ $15-18 i$ $-18-13 i$ $-13-18 i$ $-18+13 i$

15. Divide the complex numbers. Write the answer in the form $a+b i$ . $\frac{1-9 i}{2-8 i}$

Convert the complex number to polar form: $4-4 i$

Subtract. $(-8 j-6)-(-9 j+6)$

9. Simplify: $\sqrt{-50}=$ $\qquad$

$7i$ $\boxed{\phantom{7}}(\cos(\boxed{\phantom{90^\circ}})^\circ) + i \sin(\boxed{\phantom{90^\circ}})^\circ))$

Rewrite $\frac{8-i}{3-2 i}$ as $a+bi$ and find the value of $a$ (with $i=\sqrt{-1}$ ).

Rewrite the expression $\frac{8-i}{3-2 i}$ as $a+bi$ . What is the value of $a$ ? A) 2 B) $\frac{8}{3}$ C) 3 D) $\frac{11}{3}$

Number Systems Complex Numbers
11. Evaluate $1 + \frac{1+i}{\sqrt{2}} + \left(\frac{1+i}{\sqrt{2}}\right)^2 + \left(\frac{1+i}{\sqrt{2}}\right)^3 + \left(\frac{1+i}{\sqrt{2}}\right)^4$ .

54. Seja $z = \frac{i}{i+1} + \frac{1}{i}$
a) Obtenha a forma algébrica e a trigonométrica de $z$ .
b) Qual é a forma trigonométrica de $z^2$ ?

Perform the indicated operation 14.) $(1+5 i)+(6-i)$

Find the product and express it as $a + b i$ : simplify $(2 + 5 i)^{2}$ .

Find the difference and express it as $a + b i$ : $(-3 + 2 i) - (4 - 3 i)$

Evaluate $(1-2 i)(4-1 i)$ and express the result as $a + b i$ .

Simplify $i^{49}$ . What is the result? Options: $1$ , $i$ , $-i$ , $-1$ .

Divide $\frac{-3+1 i}{2 i}$ and express the result as $a + b i$ .

Final Question 2 of 45
Multiply. $(6+3 i)(6+8 i)$

Think about plotting points in the complex plane to represent the following numbers: $-3+8 i$ $4 i$ 6 5-2i
Where is each point located on the graph? $-3+8 \mathrm{i}$ is $\square$ $4 i$ is $\square$ 6 is $\square$ $5-2 i$ is $\square$ DONE

$(12-9 i)+(32-6 i)=$ $\square$ Express your answer in the form $(a+b i)$ .

Evaluate the expression $(-6+7 i)+(-8-1 i)$ and write the result in the form $a+b i$ . The real number $a$ equals $\square$ The real number $b$ equals $\square$

For numbers 3-6, simplify.
3. $\sqrt{-32}$ $32 / i$

What is the sum of the complex numbers $\sqrt{-9}$ and $\sqrt{-16}$ ? Choose from: A. $7 i$ , B. $5 i$ , C. 7, D. 5, E. -5.

2. $(7 + 3i) - (5 - 6i)$

10 Use the properties of complex numbers to simplify $(9+\sqrt{-4})+(-9-\sqrt{-16})$ . (A) $-2 i$ (B) $18-\sqrt{-20}$ (C) $18-2 i$ (D) $6 i$

9 Solve the equation $-(x-3)^{2}-3=7$ and write the answer as a complex number in the standard form $a \pm b i$ . (A) -7 (B) $-3 \pm i \sqrt{10}$ (C) $3 \pm i \sqrt{10}$ (D) $3 \pm 10 i$

Question 7 of 10 A number written in the form $a + bi$ is called a _______ number.
A. complex B. quadratic C. linear D. discriminant SUBMIT

$(4+i)(2-5 i)$

6. $(6+5i)+(6-2i)$

4. $(2 + 3i) - (5 - 2i)$ $i$ $3i$ 5 10 $15i$ $-10i$ $-9i$ $-6i^2$

b. $(-10+i)(3-9 i)$ c. $(3-4 i)-(-5-2 i)$

Find the value of $i^{99}$ .