1. Equations &amp; Inequalities

Practice this topic

Identify the real and imaginary parts of each complex number. $-4-9i$

Identify the real and imaginary parts of each complex number. $3+2i\sqrt3$

478

Write the complex number in standard form. $\frac{9+\sqrt{-16}}{3}$

Find the difference. Express your answer in standard form. $\left(2+8i\right)-\left(4-i\right)$

259

Find the sum. Express your answer in standard form. $5\left(4+7i\right)+6\left(3-2i\right)$

Perform the indicated operation. Express your answer in standard form. $\left(3+8i\right)^2$

279

Find the product. Express your answer in standard form. $2i\left(9-4i\right)\left(6+5i\right)$

Find the product of the given complex number and its conjugate. $4-5i$

269

Find the product of the given complex number and its conjugate. $-7-i$

Find the quotient. Express your answer in standard form.

$\frac{6+i}{4-2i}$

243

In Exercises 1–8, add or subtract as indicated and write the result in standard form. (7 + 2i) + (1 - 4i)

Find the quotient. Express your answer in standard form.

$\frac{-5+3i}{-7-4i}$

377

In Exercises 1–8, add or subtract as indicated and write the result in standard form. (3 + 2i) - (5 - 7i)

243

In Exercises 1–8, add or subtract as indicated and write the result in standard form. 6 - (- 5 + 4i) - (- 13 - i)

Decide whether each statement is true or false. If false, correct the right side of the equation. √-25 = 5i

268

Decide whether each statement is true or false. If false, correct the right side of the equation. i^12 = 1

245

Decide whether each statement is true or false. If false, correct the right side of the equation. (-2+7i) - (10-6i)= -12+i

289

In Exercises 9–20, find each product and write the result in standard form. (7 - 5i)(- 2 - 3i)

250

In Exercises 9–20, find each product and write the result in standard form. (3 + 5i)(3 - 5i)

235

In Exercises 9–20, find each product and write the result in standard form. (- 5 + i)(- 5 - i)

227

Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) √24

273

Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) √-36

326

Write each number as the product of a real number and i. √-25

204

Perform each operation. Write answers in standard form. 15i- (3+2i) -11

230

In Exercises 21–28, divide and express the result in standard form. 2i/(1 + i)

257

Perform each operation. Write answers in standard form. (-8+2i)(-1+i)

In Exercises 21–28, divide and express the result in standard form. 5i/(2 - i)

269

Perform each operation. Write answers in standard form. (5-11i)(5+11i)

In Exercises 21–28, divide and express the result in standard form. 8i/(4 - 3i)

288

Find each product or quotient. Simplify the answers. √-13 * √-13

224

Find each product or quotient. Simplify the answers. √-3 * √-8

212

In Exercises 29–36, simplify and write the result in standard form. √-108

312

Find each product or quotient. Simplify the answers. √-30 / √-10

In Exercises 29–36, simplify and write the result in standard form. √(3^2 - 4 × 2 × 5)

246

Simplify each power of i. i^1001

340

In Exercises 29–36, simplify and write the result in standard form. √(1^2 - 4 × 0.5 × 5)

246

Simplify each power of i. i^-27

215

Simplify each power of i. 1/i^17

215

Write each number in standard form a+bi. -6-√-24 / 2

221

Write each number in standard form a+bi. 10+ √-200 / 5

214

In Exercises 37–52, perform the indicated operations and write the result in standard form. (- 3 - √-7)^2

245

Write each number in standard form a+bi. -3+ √-18 / 24

323

In Exercises 37–52, perform the indicated operations and write the result in standard form. (- 8 + √-32)/24

386

In Exercises 37–52, perform the indicated operations and write the result in standard form. (- 6 - √-12)/48

Find each sum or difference. Write answers in standard form. (2-5i) - (3+4i) - (-2+i)

451

Find each sum or difference. Write answers in standard form. (-4-i) - (2+3i) + (-4+5i)

224

Find each sum or difference. Write answers in standard form. 3√7 - (4√7-i) -4i + (-2√7+5i)

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i^44

244

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i^114

225

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i^135

Find each product. Write answers in standard form. (3+i)(3-i)

206

Find each product. Write answers in standard form. (-2-3i)(-2+3i)

235

Find each product. Write answers in standard form. (√6+i)(√6-i)

215

In Exercises 65–70, perform the indicated operation(s) and write the result in standard form. (2 - 3i)(1 - i) - (3 - i)(3 + i)

243

In Exercises 65–70, perform the indicated operation(s) and write the result in standard form. (8 + 9i)(2 - i) - (1 - i)(1 + i)

249

In Exercises 65–70, perform the indicated operation(s) and write the result in standard form. (2 + i)^2 - (3 - i)^2

258

Find each product. Write answers in standard form. (3-i)(3+1)(2-6i)

292

Evaluate (x^2 + 19)/(2 - x) for x = 3i.

274

Find each quotient. Write answers in standard form. 2-i / 2+i

517

Find each quotient. Write answers in standard form. 1-3i / 1+i

282

Simplify each power of i. i^25

214

Simplify each power of i. i^29

205

Simplify each power of i. i^26

232

In Exercises 95–99, perform the indicated operations and write the result in standard form. 4/(2 + i)(3 - i)

414

In Exercises 95–99, perform the indicated operations and write the result in standard form. (1 + i)/(1 + 2i) + (1 - i)/(1 - 2i)

241

Simplify each power of i. i^-14

233

Simplify each power of i. 1/i^-11

217

Simplify each power of i. 1/i^-12

210