Exam Study

Radical Expressions and Complex Numbers

Closure of Sets Under the Four Basic Operations

• Real Numbers are closed under all operations.• Irrational Numbers are not closed under the

four basic operations. • Rational Numbers are closed under the four

basic operations. • Integers are closed on all except division.

Radicals and Rational Exponents

Simplifying Radicals and Rational Exponents

• Examples:

Adding and Subtracting Radicals

1. Simplify all Radicals2. Identify radicals with the SAME INDEX and

SAME RADICAND. (only these can be combined)

3. For common radicals. Add/subtract the coefficients and KEEP THE COMMON RADICAL

Example:

3√−40𝑎7+2𝑎2 3√135𝑎4

Example

√98 𝑥4 𝑦 2−3 𝑥2 𝑦 √2

Multiplying Radicals

1. Multiply the coefficients, then use the PRODUCT RULE:

2. SIMPLIFY the resulting radical

Example

24√𝑝2𝑞 ∙7 4√𝑝3𝑞10

Example

(√𝑥−√9 ) (√𝑥+9 )

Steps to Divide Radicals

Example:

Example:

plex Numbers

• and are REAL numbers.

• Each term has a name: = real part, = imaginary part

• When the complex number is simply a REAL number• When is an imaginary number• When , then that is called pure imaginary number

Complex Numbers

I

Pure Imaginary Numbers

Imaginary NumbersReal Numbers

What is the definition of a complex number?

a. A number of the form where and are real.

b. A number of the form where and is real.

c. A number of the form where is real and .

d. A number of the form where is real, and .

Powers of Power Answer

Adding and Subtracting Complex Numbers

(8+3 𝑖 )+(7+5 𝑖 )

Adding and Subtracting Complex Numbers

(8+3 𝑖 )− (7−3 𝑖 )

Multiplying Complex Numbers

(8+12 𝑖)(4−2𝑖)

Identify: Real or Complex