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Introduction to Simplifying Radicals
Simplify Each Radical:
1) √48
2) √128
5) √25x2
6) √72x8
9) √x9
10) √x9y10
13) √162x10y5
14) √75x7y3
17) √108x16y25
18) √98x1000y500
3) √363
7) √432x16y8
11) √x9y11
15) √300x5y12
19) √600x11y14
4) √45
8) √392x100y210
12) √25x9y11
16) √169x100y64
20) √-36
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Imaginary Numbers
You can’t take the square root of -36 (or of any other negative number). Think about it.
36 = ± 6, because 6 · 6 = 36 and -6 · -6 = 36. But you cannot multiply a number by itself and get
a negative number. We use the imaginary unit i to write the square root of any negative number.
√-1 = i
√-36

√36 · -1

Simplify Each Radical:
1) √-8
2) √-50
6) √-245
5) 2√-384
6i
3) 4√-242
4) 7√-125
7) √-588
8) √-361
Simplifying with i:
i1 =
i5 =
i9 =
i13 =
i33 =
i2 =
i6 =
i10 =
i14 =
i38 =
i3 =
i7 =
i11 =
i15 =
i43 =
i4 =
i8 =
i12 =
i16 =
i44 =
When simplifying and using operations with imaginary numbers, you are not allowed to leave i
with an exponent. How possible terms can you have when you simplify?
We have a simple way of remembering how to simplify:
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1) i1
2) i2
3) i3
4) i4
5) i5
6) i6
7) i7
8) i8
9) i9
10) i10
11) i11
12) i12
13) i21
14) i33
15) i32
16) i26
17) (√-10)2
18) √-10 · √-20
19) √-3 · √-12
20) √-18 · √-6
21) (3i)2
22) (i√3 )2
23) (-i )2
24) – i2
25) (9i5)(-6i13)
26) (-8i7)(-13i17)
27) (-5i11)(12i10)
28) (-11i23)(-12i16)
Complex Numbers
A complex number is a number that is the sum of a real number and a regular number. Each
complex number should be written in the standard form a + bi.
Example:
8 + 3i
Perform the indicated operation:
29) (11- 5i)(7 – 3i)
30) (4 + 5i)(7 – 3i)
31) 2i2(3 – 8i) – 4i(12 – 7i)
32) (9 + 3i)(12 + 2i)
33) 5i2(3 + 2i) – 3i2(4 + 8i)
34) (4 + 2i) + (7 – 2i)
35) 3(6-2i) – 4(4 + 3i)
36) 8i(9 + 2i) – 3i(3 - 7i)
37) (3 – 2i)(4 + 5i)
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Practice Problems: #1 can be done on the sheet, the rest should be done in your NB.
1) Simplify:
i21 =
i58 =
i92 =
i123 =
i45 =
i26 =
i61 =
i19 =
i129 =
i58 =
i35 =
i72 =
i14 =
i106 =
i74 =
i48 =
i87 =
i66 =
i116 =
i1,000 =
2) √-108
3) √-486
4) 8√-392
5) -7√-675
6) 9√-441
7) 12(5 – 7i) + 10(-6 + 8i)
8) 3i(4 – 10i) – 7(8 + 5i)
9) 6(3 + 2i) + 8i(4 + 9i)
10) 9i(4 – 11i) – 6i2(-10 – 4i)
11) (3 – 2i)(4 + 5i)
12) (10- 4i)(6 – 2i)
13) (9 + i)(8 – 5i)
14) (6 + 7i)(6 + 2i)
15) (5 – 8i)(11 + 9i)
16) (10- 3i)(7 – 12i)