M
Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 2

Write Numbers with Imaginary Unit

The imaginary unit is i = square root of −1. Rewrite square roots of negatives using i, then simplify.

Jump to section:

Key Ideas Easy Medium Hard Hard 2 Hard 3 Hard 4

i = square root of −1

i² = −1

What to Remember

The imaginary unit

i is defined as the square root of −1.

i = square root of −1

That means i² = −1.

Pull out negative in radical

square root of −5 = isquare root of 5

square root of −36 = isquare root of 36 = 6i

square root of −12 = isquare root of 12 = 2isquare root of 3

Level 1

Easy

Pull i out. Leave the leftover under the radical.

Example
Write using i.
square root of −6

Pull the negative out of the radical

square root of −6 = isquare root of 6

Answer: isquare root of 6

1
Write using i.
square root of −11
Show Answer

Pull the negative out of the radical

square root of −11 = isquare root of 11

Answer: isquare root of 11

2
Write using i.
square root of −2
Show Answer

Pull the negative out of the radical

square root of −2 = isquare root of 2

Answer: isquare root of 2

3
Write using i.
square root of −13
Show Answer

Pull the negative out of the radical

square root of −13 = isquare root of 13

Answer: isquare root of 13

4
Write using i.
square root of −10
Show Answer

Pull the negative out of the radical

square root of −10 = isquare root of 10

Answer: isquare root of 10

5
Write using i.
square root of −1
Show Answer

Pull the negative out of the radical

square root of −1 = i

Answer: i

Level 2

Medium

If the leftover is a perfect square, write a multiple of i. The last two have a minus in front.

Example
Write using i.
square root of −25

Pull the negative out of the radical

square root of −25 = isquare root of 25

square root of 25 = 5

i · 5

5i

Answer: 5i

1
Write using i.
square root of −9
Show Answer

Pull the negative out of the radical

square root of −9 = isquare root of 9

square root of 9 = 3

i · 3

3i

Answer: 3i

2
Write using i.
square root of −64
Show Answer

Pull the negative out of the radical

square root of −64 = isquare root of 64

square root of 64 = 8

i · 8

8i

Answer: 8i

3
Write using i.
square root of −100
Show Answer

Pull the negative out of the radical

square root of −100 = isquare root of 100

square root of 100 = 10

i · 10

10i

Answer: 10i

4
Write using i.
−square root of −16
Show Answer

Pull the negative out of the radical

−square root of −16 = −isquare root of 16

square root of 16 = 4

−i · 4

−4i

Answer: −4i

5
Write using i.
−square root of −7
Show Answer

Pull the negative out of the radical

−square root of −7 = −isquare root of 7

Answer: −isquare root of 7

Level 3

Hard

Pull i out, then pull a perfect square out of what is left.

Example
Write using i.
square root of −12

Pull the negative out of the radical

square root of −12 = isquare root of 12

isquare root of 12 = isquare root of 2 · 2 · 3

i · 2 · square root of 3

2isquare root of 3

Answer: 2isquare root of 3

1
Write using i.
square root of −50
Show Answer

Pull the negative out of the radical

square root of −50 = isquare root of 50

isquare root of 50 = isquare root of 2 · 5 · 5

i · 5 · square root of 2

5isquare root of 2

Answer: 5isquare root of 2

2
Write using i.
−square root of −18
Show Answer

Pull the negative out of the radical

−square root of −18 = −isquare root of 18

−isquare root of 18 = −isquare root of 2 · 3 · 3

−i · 3 · square root of 2

−3isquare root of 2

Answer: −3isquare root of 2

3
Write using i.
2square root of −20
Show Answer

Pull the negative out of the radical

2square root of −20 = 2isquare root of 20

2isquare root of 20 = 2isquare root of 2 · 2 · 5

2i · 2 · square root of 5

4isquare root of 5

Answer: 4isquare root of 5

4
Write using i.
square root of −27
Show Answer

Pull the negative out of the radical

square root of −27 = isquare root of 27

isquare root of 27 = isquare root of 3 · 3 · 3

i · 3 · square root of 3

3isquare root of 3

Answer: 3isquare root of 3

5
Write using i.
−3square root of −8
Show Answer

Pull the negative out of the radical

−3square root of −8 = −3isquare root of 8

−3isquare root of 8 = −3isquare root of 2 · 2 · 2

−3i · 2 · square root of 2

−6isquare root of 2

Answer: −6isquare root of 2

Level 4

Hard 2

Rewrite each square root. Combine like terms.

Example
Simplify. square root of −36 + 3square root of 9 − 2square root of −4 + square root of 16

square root of −36 = 6i

3square root of 9 = 3 · 3 = 9

2square root of −4 = 2 · 2i = 4i

square root of 16 = 4

6i + 9 − 4i + 4

13 + 2i

Answer: 13 + 2i

1
Simplify. 2square root of 49 −square root of −16 + square root of −9 − 3square root of 4
Show Answer

2square root of 49 = 14

square root of −16 = 4i

square root of −9 = 3i

3square root of 4 = 6

14 − 4i + 3i − 6

8 − i

Answer: 8 − i

2
Simplify. 4square root of 9 + 2square root of −25 −square root of 36 −square root of −1
Show Answer

4square root of 9 = 12

2square root of −25 = 10i

square root of 36 = 6

square root of −1 = i

12 + 10i − 6 − i

6 + 9i

Answer: 6 + 9i

3
Simplify. square root of −81 − 2square root of 16 + 5square root of −4 + square root of 1
Show Answer

square root of −81 = 9i

2square root of 16 = 8

5square root of −4 = 10i

square root of 1 = 1

9i − 8 + 10i + 1

−7 + 19i

Answer: −7 + 19i

4
Simplify. 3square root of 64 − 4square root of −9 + square root of −36 −square root of 25
Show Answer

3square root of 64 = 24

4square root of −9 = 12i

square root of −36 = 6i

square root of 25 = 5

24 − 12i + 6i − 5

19 − 6i

Answer: 19 − 6i

5
Simplify. square root of 100 + square root of −49 − 2square root of 36 − 3square root of −4
Show Answer

square root of 100 = 10

square root of −49 = 7i

2square root of 36 = 12

3square root of −4 = 6i

10 + 7i − 12 − 6i

−2 + i

Answer: −2 + i

Level 5

Hard 3

Multiply. Convert negative radicands to i first.

Example
Simplify. square root of −5 · square root of −20

Pull the negative out of the radical

square root of −5 = isquare root of 5

square root of −20 = isquare root of 20

square root of 20 = 2square root of 5

square root of −20 = 2isquare root of 5

(isquare root of 5)(2isquare root of 5)

2i² · 5

2(−1)(5)

−10

Answer: −10

1
Simplify. square root of −8 · square root of −2
Show Answer

Pull the negative out of the radical

square root of −8 = isquare root of 8

square root of 8 = 2square root of 2

square root of −8 = 2isquare root of 2

square root of −2 = isquare root of 2

(2isquare root of 2)(isquare root of 2)

2i² · 2

2(−1)(2)

−4

Answer: −4

2
Simplify. 3square root of −3 · 2square root of −12
Show Answer

Pull the negative out of the radical

3square root of −3 = 3isquare root of 3

2square root of −12 = 2isquare root of 12

square root of 12 = 2square root of 3

2square root of −12 = 4isquare root of 3

(3isquare root of 3)(4isquare root of 3)

12i² · 3

12(−1)(3)

−36

Answer: −36

3
Simplify. 3isquare root of 10 · −2isquare root of −40
Show Answer

Pull the negative out of the radical

square root of −40 = isquare root of 40

square root of 40 = 2square root of 10

square root of −40 = 2isquare root of 10

−2i · 2isquare root of 10

−4i²square root of 10

−4(−1)square root of 10

4square root of 10

3isquare root of 10 · 4square root of 10

12i · 10

120i

Answer: 120i

4
Simplify. −3isquare root of −20 · −2isquare root of −45
Show Answer

Pull the negative out of the radical

square root of −20 = isquare root of 20

square root of 20 = 2square root of 5

square root of −20 = 2isquare root of 5

−3i · 2isquare root of 5

−6i²square root of 5

−6(−1)square root of 5

6square root of 5

square root of −45 = isquare root of 45

square root of 45 = 3square root of 5

square root of −45 = 3isquare root of 5

−2i · 3isquare root of 5

−6i²square root of 5

−6(−1)square root of 5

6square root of 5

6square root of 5 · 6square root of 5

36 · 5

180

Answer: 180

5
Simplify. 2isquare root of 15 · 3isquare root of −60
Show Answer

Pull the negative out of the radical

square root of −60 = isquare root of 60

square root of 60 = 2square root of 15

square root of −60 = 2isquare root of 15

2isquare root of 15 · 3i · 2isquare root of 15

12i³ · 15

i³ = −i

12(−i)(15)

−180i

Answer: −180i

6
Simplify. 4isquare root of −18 · −3isquare root of −8
Show Answer

Pull the negative out of the radical

square root of −18 = isquare root of 18

square root of 18 = 3square root of 2

square root of −18 = 3isquare root of 2

4i · 3isquare root of 2

12i²square root of 2

12(−1)square root of 2

−12square root of 2

square root of −8 = isquare root of 8

square root of 8 = 2square root of 2

square root of −8 = 2isquare root of 2

−3i · 2isquare root of 2

−6i²square root of 2

−6(−1)square root of 2

6square root of 2

(−12square root of 2)(6square root of 2)

−72 · 2

−144

Answer: −144

Level 6

Hard 4

Write a squared factor twice. Rearrange with the commutative property of multiplication.

Example
Simplify. 2i(5i)²(−3i)

Write (5i) twice because there are two copies of 5i.

(2i)(5i)(5i)(−3i)

Use the commutative property of multiplication to rearrange the factors.

2 · 5 · 5 · (−3) · i · i · i · i

2 · 5 · 5 · (−3) = −150

i · i · i · i = i² · i² = (−1) · (−1) = 1

−150 · 1 = −150

Answer: −150

1
Simplify. (3i)²(4i)(−i)
Show Answer

Write (3i) twice because there are two copies of 3i.

(3i)(3i)(4i)(−i)

Use the commutative property of multiplication to rearrange the factors.

3 · 3 · 4 · (−1) · i · i · i · i

3 · 3 · 4 · (−1) = −36

i · i · i · i = i² · i² = (−1) · (−1) = 1

−36 · 1 = −36

Answer: −36

2
Simplify. 6i(−2i)²(i)
Show Answer

Write (−2i) twice because there are two copies of −2i.

(6i)(−2i)(−2i)(i)

Use the commutative property of multiplication to rearrange the factors.

6 · (−2) · (−2) · 1 · i · i · i · i

6 · (−2) · (−2) · 1 = 24

i · i · i · i = i² · i² = (−1) · (−1) = 1

24 · 1 = 24

Answer: 24

3
Simplify. (−4i)(3i)(2i)²
Show Answer

Write (2i) twice because there are two copies of 2i.

(−4i)(3i)(2i)(2i)

Use the commutative property of multiplication to rearrange the factors.

(−4) · 3 · 2 · 2 · i · i · i · i

(−4) · 3 · 2 · 2 = −48

i · i · i · i = i² · i² = (−1) · (−1) = 1

−48 · 1 = −48

Answer: −48

4
Simplify. (7i)²(−i)(2i)
Show Answer

Write (7i) twice because there are two copies of 7i.

(7i)(7i)(−i)(2i)

Use the commutative property of multiplication to rearrange the factors.

7 · 7 · (−1) · 2 · i · i · i · i

7 · 7 · (−1) · 2 = −98

i · i · i · i = i² · i² = (−1) · (−1) = 1

−98 · 1 = −98

Answer: −98

5
Simplify. (−6i)(−2i)(3i)²
Show Answer

Write (3i) twice because there are two copies of 3i.

(−6i)(−2i)(3i)(3i)

Use the commutative property of multiplication to rearrange the factors.

(−6) · (−2) · 3 · 3 · i · i · i · i

(−6) · (−2) · 3 · 3 = 108

i · i · i · i = i² · i² = (−1) · (−1) = 1

108 · 1 = 108

Answer: 108

Need help writing numbers with i?

Book a Session with Amy