H Algebra 2/Trig · Unit 2
The imaginary unit is i = √(−1). Rewrite square roots of negatives using i, then simplify.
The imaginary unit
i is defined as the square root of −1.
i = √(−1)
That means i² = −1.
Pull out negative in radical
√(−5) = i√5
√(−36) = i√36 = 6i
√(−12) = i√12 = 2i√3
Pull i out. Leave the leftover under the radical.
Pull the negative out of the radical
√(−6) = i√6
Answer: i√6
Pull the negative out of the radical
√(−11) = i√11
Answer: i√11
Pull the negative out of the radical
√(−2) = i√2
Answer: i√2
Pull the negative out of the radical
√(−13) = i√13
Answer: i√13
Pull the negative out of the radical
√(−10) = i√10
Answer: i√10
Pull the negative out of the radical
√(−1) = i
Answer: i
If the leftover is a perfect square, write a multiple of i. The last two have a minus in front.
Pull the negative out of the radical
√(−25) = i√25
√25 = 5
i · 5
5i
Answer: 5i
Pull the negative out of the radical
√(−9) = i√9
√9 = 3
i · 3
3i
Answer: 3i
Pull the negative out of the radical
√(−64) = i√64
√64 = 8
i · 8
8i
Answer: 8i
Pull the negative out of the radical
√(−100) = i√100
√100 = 10
i · 10
10i
Answer: 10i
Pull the negative out of the radical
−√(−16) = −i√16
√16 = 4
−i · 4
−4i
Answer: −4i
Pull the negative out of the radical
−√(−7) = −i√7
Answer: −i√7
Pull i out, then pull a perfect square out of what is left.
Pull the negative out of the radical
√(−12) = i√12
i√12 = i√(2 · 2 · 3)
i · 2 · √3
2i√3
Answer: 2i√3
Pull the negative out of the radical
√(−50) = i√50
i√50 = i√(2 · 5 · 5)
i · 5 · √2
5i√2
Answer: 5i√2
Pull the negative out of the radical
−√(−18) = −i√18
−i√18 = −i√(2 · 3 · 3)
−i · 3 · √2
−3i√2
Answer: −3i√2
Pull the negative out of the radical
2√(−20) = 2i√20
2i√20 = 2i√(2 · 2 · 5)
2i · 2 · √5
4i√5
Answer: 4i√5
Pull the negative out of the radical
√(−27) = i√27
i√27 = i√(3 · 3 · 3)
i · 3 · √3
3i√3
Answer: 3i√3
Pull the negative out of the radical
−3√(−8) = −3i√8
−3i√8 = −3i√(2 · 2 · 2)
−3i · 2 · √2
−6i√2
Answer: −6i√2
Rewrite each square root. Combine like terms.
√(−36) = 6i
3√9 = 3 · 3 = 9
2√(−4) = 2 · 2i = 4i
√16 = 4
6i + 9 − 4i + 4
13 + 2i
Answer: 13 + 2i
2√49 = 14
√(−16) = 4i
√(−9) = 3i
3√4 = 6
14 − 4i + 3i − 6
8 − i
Answer: 8 − i
4√9 = 12
2√(−25) = 10i
√36 = 6
√(−1) = i
12 + 10i − 6 − i
6 + 9i
Answer: 6 + 9i
√(−81) = 9i
2√16 = 8
5√(−4) = 10i
√1 = 1
9i − 8 + 10i + 1
−7 + 19i
Answer: −7 + 19i
3√64 = 24
4√(−9) = 12i
√(−36) = 6i
√25 = 5
24 − 12i + 6i − 5
19 − 6i
Answer: 19 − 6i
√100 = 10
√(−49) = 7i
2√36 = 12
3√(−4) = 6i
10 + 7i − 12 − 6i
−2 + i
Answer: −2 + i
Multiply. Convert negative radicands to i first.
Pull the negative out of the radical
√(−5) = i√5
√(−20) = i√20
√20 = 2√5
√(−20) = 2i√5
(i√5)(2i√5)
2i² · 5
2(−1)(5)
−10
Answer: −10
Pull the negative out of the radical
√(−8) = i√8
√8 = 2√2
√(−8) = 2i√2
√(−2) = i√2
(2i√2)(i√2)
2i² · 2
2(−1)(2)
−4
Answer: −4
Pull the negative out of the radical
3√(−3) = 3i√3
2√(−12) = 2i√12
√12 = 2√3
2√(−12) = 4i√3
(3i√3)(4i√3)
12i² · 3
12(−1)(3)
−36
Answer: −36
Pull the negative out of the radical
√(−40) = i√40
√40 = 2√10
√(−40) = 2i√10
−2i · 2i√10
−4i²√10
−4(−1)√10
4√10
3i√10 · 4√10
12i · 10
120i
Answer: 120i
Pull the negative out of the radical
√(−20) = i√20
√20 = 2√5
√(−20) = 2i√5
−3i · 2i√5
−6i²√5
−6(−1)√5
6√5
√(−45) = i√45
√45 = 3√5
√(−45) = 3i√5
−2i · 3i√5
−6i²√5
−6(−1)√5
6√5
6√5 · 6√5
36 · 5
180
Answer: 180
Pull the negative out of the radical
√(−60) = i√60
√60 = 2√15
√(−60) = 2i√15
2i√15 · 3i · 2i√15
12i³ · 15
i³ = −i
12(−i)(15)
−180i
Answer: −180i
Pull the negative out of the radical
√(−18) = i√18
√18 = 3√2
√(−18) = 3i√2
4i · 3i√2
12i²√2
12(−1)√2
−12√2
√(−8) = i√8
√8 = 2√2
√(−8) = 2i√2
−3i · 2i√2
−6i²√2
−6(−1)√2
6√2
(−12√2)(6√2)
−72 · 2
−144
Answer: −144
Write a squared factor twice. Rearrange with the commutative property of multiplication.
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
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
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
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
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
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
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