Simplify Each Radical:



Introduction to Simplifying Radicals

Simplify Each Radical:

1) √48                          2) √128                        3) √363                        4) √45

 

 5) √25x2                       6) √72x8                       7) √432x16y8                8) √392x100y210

 

 

9) √x9                           10) √x9y10                           11) √x9y11                    12) √25x9y11

 

 

13) √162x10y5                   14) √75x7y3                       15) √300x5y12                   16) √169x100y64

 

 

17) √108x16y25                 18) √98x1000y500               19) √600x11y14                 20) √-36

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 ( 6i

Simplify Each Radical:

1) √-8 2) √-50 3) 4√-242 4) 7√-125

5) 2√-384 6) √-245 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)

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)

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