Essential Question: How do we simplify square roots of negative numbers?

5-6: Complex NumbersOperations with Radicals

Simplifying a radical: Option #1 Break a number into prime factors. Pull any pairs out as one number outside the radical Multiply any numbers remaining inside & outside the radical

Example #1 75 = 5 • 5 • 3 =

Example #2 96 = 2 • 2 • 2 • 2 • 2 • 3 = 2 • 2

755 3

962 3 4 6

5-6: Complex NumbersOperations with Radicals

Simplifying a radical: Option #2 Divide by perfect squares. A perfect square is any number times itself:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, etc. Simplify all portions of the radical

Example #1

Example #2

75 25 3 25 3 5 3

96 16 6 16 6 4 6

5-6: Complex NumbersSome rules with radicals

When numbers INSIDE a radical match, the numbers OUSTIDE can be added/subtracted (rule of like terms)

Sometimes, radicals must be simplified before they can be combined Example: 18 50 27

9 2 25 2 9 3

3 2 5 2 3 3

2 2 3 3

5-6: Complex NumbersSome rules with radicals (continued)

You shouldn’t leave a radical in the denominator of a fraction

To remove it, we rationalize the denominator. Multiply the top and bottom of the fraction by the radical in the denominator.

Examples:

3

3

5 5 3

33

2 5 2 5 2 5 2 10 10

6 3 2 33 18 3 2

2

2

5-6: Complex NumbersAssignment

Page 8832 – 30, evens

Essential Question: How do we simplify square roots of negative numbers?

5-6: Complex NumbersThe imaginary number i is defined as the

number whose square is -1.So i2 = -1, and

To simplify square roots of negative numbersTake the negative sign outside the square root,

replace it with i.Simplify the number underneath the square

root as normal. Numbers outside the square root come before the i.

Example:

1i

8 4 28 2 2i i i

5-6: Complex NumbersSolve

2

12

36

2i

12 4 3 2 3i i i

36 6i i

5-6: Complex NumbersImaginary numbers and real numbers make

up the set of complex numbers.Complex numbers are written in the form a +

biThat means the real number gets written first,

followed by the imaginary number.Example:

Write the complex number in the form a + bi

9 6

6 3

9 6 9 6

3 6

i

i

i

5-6: Complex NumbersWrite the complex number in a + bi

form.

18 7 18 7

18 7i

9 2 7i

3 2 7i

7 3 2i

5-6: Complex NumbersYou can apply real number concepts to

complex numbers.Complex numbers have additive inverses (or

“opposites”)It’s simply the opposite of the real number

added to the opposite of the imaginary numberExample: Find the additive inverse of -2 + 5i.

The opposite of -2 is 2The opposite of 5i is -5i.So the additive inverse of -2 + 5i is 2 – 5i.

5-6: Complex NumbersFind the additive inverse of each number

5i

4 3i

a bi

5i

4 3i

a bi

5-6: Complex NumbersAssignment

Page 274Problems 1 – 18 and 24 – 28All problems

Essential Question: How do we simplify square roots of negative numbers?

5-6: Complex NumbersAdding & Subtracting Complex Numbers

Simply combine the real parts with the imaginary parts

Example(5 + 7i) + (-2 + 6i)5 + -2 + 7i + 6i3 + 13i

5-6: Complex NumbersSimplify each expression

(4 6 ) 3i i

(8 3 ) (2 4 )i i

7 (3 2 )i

4 3i

6 i

4 2i

5-6: Complex NumbersMultiplying Complex Numbers

If i = , then i2 = -1Example

(5i)(-4i)-20i2

Replace i2 with -1-20(-1)20

1

5-6: Complex NumbersSimplify the expression

(12 )(7 )i i

8484( 1)

284i

5-6: Complex NumbersMultiplying Complex NumbersFOIL Example

(2 + 3i)(-3 + 5i)-6 + 10i – 9i + 15i2 Combine like terms-6 + i + 15(-1) Replace i2 with -1-6 + i – 15 Combine like terms again-21 + i

5-6: Complex NumbersSimplify the expression

(6 5 )(4 3 )i i 224 18 20 15i i i

24 38 15( 1)i 24 38 15i 9 38i

5-6: Complex NumbersSimplify the expression

(4 9 )(4 3 )i i 216 12 36 27i i i

16 24 27( 1)i 16 24 27i 43 24i

5-6: Complex NumbersAssignment

Page 274Problems 29-40 (all) and 58-66 (even)