Radicals and Rational Exponents

Chapter 7, Section 3: Radicals and Rational Exponents

Suppose that m and n are integers and that there are no zero divisors. Complete
the following:

Calculator Activity: Use your calculator to evaluate each of the following.
Compare

the answers.

If n is a natural number greater than 1 and is a real number, then

Question: When is not a real number?

Rewrite each of the following in radical form and simplify:

If m and n are positive integers and the GCF (m,n)=1 and x>0,
then

EX8. Evaluate in 2 different ways.

1)
2)

EX9. Evaluate in 2 different ways

1)
2)

Changing from radical to exponential form and vice versa:

Simplifying Expressions Involving Rational Exponents:

Simplifying Radical Expressions by Reducing the Index

Chapter 7, Section 6: Radical Equations

EX1. Solve: by inspection; i.e., just determine logically what
number or numbers will make this equation a true statement.

Solve each of the following equations and check your answers.

Question: Why is it important to check your answers when solving radical
equations?

Note: When solving an equation containing 2 or more radical terms, you need to
remember that

EX8. Find the value of r when n = 5, V = $1338.25, and P = $1000.

Chapter 7, Section 7: Complex Numbers

EX1. Solve:

EX2. Solve:

A complex number is any number of the form a+bi where a and b are real
numbers and . Note: .

Adding and Subtracting Complex Numbers

EX3. ( 2 + 3i ) + ( 4 - i ) =
EX4. ( 5 - 2i ) + ( -3 - 4i ) =
EX5. ( 2 + 3i ) - ( 4 - i ) =
EX6. ( 5 - 2i ) - ( -3 - 4i ) =

Multiplying Complex Numbers

EX7. ( 2 + 3i )( 4 - i ) =
EX8. ( 5 - 2i )( -3 - 4i ) =

The conjugate of the complex number . This is denoted by
.

Dividing Complex Numbers: [Note–To divide complex numbers, multiply both
the numerator and the denominator by the conjugate of the denominator.]

Simplifying the expression where is an integer:

Do you observe any pattern? If so, what is it?

Divisibility Rule for 4: A number N is divisible by 4 if and only if the
number
represented by the last 2 digits of N is divisible by 4.

EX11. Is 3724 divisible by 4? Why?
EX12. Is 5817 divisible by 4? Why?
EX13. Simplify:
EX14. Simplify:
EX15. Suppose that , where k is the units digit in the exponent. What digit(s) could k be?

Chapter 8, Section 1: Solving Quadratic Equations by Completing the Square

Solve:  by factoring.
Solve: by factoring
How is this problem different from the first problem?

The Square Root Property: If

Note: If , then there are ___________________________________.
If , then there is ____________________________________.
If , then there are ____________________________________.

EX1. Solve:

EX2. Solve:

A demonstration of the geometry associated with completing the square.

What number must be added to each of the following to complete the square?
Write each perfect square trinomial as a binomial squared.

For each of the following, determine a value for k that would make the trinomial a
perfect square trinomial.

Solve each of the following by the process of completing the square:

Chapter 8, Section 2: Solving Quadratic Equations by the Quadratic Formula

EX1. Solve: by completing the square

EX2. Solve: (The General Quadratic Equation in One
Variable) by completing the square.

The quadratic formula is:

The radicand, in the quadratic formula is called the discriminant.
If , there are ________________________________________.
If , there is _________________________________________ .
If , there are _________________________________________.

Use the quadratic formula to solve each of the following:

EX6. The product of two consecutive odd negative integers is 143. Find the
integers.

EX7. A rectangle is 2 inches longer than it is wide. Numerically, its area exceeds
its perimeter by 11. Find the perimeter of the rectangle.