Unit 2A: Simplifying Radicals and Imaginary/Complex Numbers

Honors Algebra 2/Trig

Unit Goals

▪ Students will be able to:

▪ Simplify Radicals of varying index

▪ Add, Subtract, Multiply, and Divide Radicals

▪ Rationalize the denominator when presented with

radicals and radical expressions

▪ Identify and simplify complex and imaginary numbers

Day 1 – Simplifying RadicalsBook Section 5.3

Warm Up

▪ Simplify the following radicals:

▪ 25 =

▪ 64 =

▪ 600 =

Reducing Radicals

▪ Method 1: Perfect Factors

1. “Break down” (a.k.a. factor) the number

inside the radical, using the largest “perfect”

factor.

2. Rewrite the radical as a product of two factors.

3. Cross out the perfect root and write what it

equals in front of the radical.

4. Bring down any “left-overs,” and write them

with a multiplication sign between them.

5. Multiply the numbers in front of the radical.

Reducing Radicals

▪ Method 2: Groups

1. “Break down” (a.k.a. factor) the number

inside the radical. Write out the prime

factorization

2. Determine the group amount.

3. Circle all groups (if any).

4. Write one number from each group outside of

the radical. Multiply these numbers.

5. Leave any “left-over” numbers inside the

radical and multiply.

Examples:

1. 600 2. 72

3.316 4.

332

Examples:

5.416 6.

760

7.4810𝑥8𝑦2𝑧5 8.

596𝑥3𝑦7𝑧15

Properties of Radicals

▪ Product Property

▪ 𝑎 × 𝑏 = 𝑎𝑏

▪ Just multiply the numbers underneath the radical!

▪ Simplify radicals either before OR after multiplying.

▪ If there are numbers out front, multiply them first.

▪ Quotient Property

▪𝑎

𝑏=

𝑎

𝑏

▪ A fraction underneath a radical can be broken up into a

fraction of radicals.

▪ We cannot have radicals left in the denominator. To

rationalize the denominator, multiply the numerator and

denominator by the radical in the denominator (for

square roots only)

Product Property Examples:

1. 3 ∙ 5 2. 10 ∙ 14

3.237 ∙

314 4.

48 ∙

410

Quotient Property Examples:

1.36

252.

75

4

3.3

104.

12

7

Quotient Property Examples (2):

5.3 27

86.

4 32

5

7.3 35

128.

48

4

Adding and Subtracting

Radicals

▪ To add and subtract radicals, treat the radicals like

variables!

▪ We can only combine like terms.

▪ The type of radical AND the number inside the radical

MUST be the same in order to add/subtract.

▪ It is possible that we cannot add or subtract radicals.

▪ You cannot add or subtract two radicals with different

indexes.

Adding and Subtracting Examples

1. 3 + 3

2. 5 + 8 5

3. 6 7 − 8 7

4. 2 + 3

5. 237 + 2

37

6. 348 −

48

7. 635 + 8 5

8. 5316 + 7

354

Page 267, #19-47 1st Column Page 411, #34-46 1st Column

#50, 52, 54

Homework