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Exponents and Radicals
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Exponents and Radicals
RADICALS & RATIONAL EXPONENTS
Radicals and Rational Exponents
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RADICAL EQUATION
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Solving Radical Equations
Exponents and Radicals
Exponents and Radicals
Roots;Rational Exponents;Radical Equations
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Solving Radical Equations
Radicals and Rational Exponents
exponential_and_radical_properties
Roots, Radicals, and Root Functions
Multiplication of Radicals
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SOLVING RADICAL EQUATIONS
Equations Containing Radicals and Complex Numbers
Square Roots and Radicals
Solving Radical Equations in One Variable Algebraically
Polynomials and Radicals
Roots,Radicals,and Fractional Exponents
Adding, Subtracting, and Multiplying Radical Expressions
Square Formula and Powers with Radicals
Simplifying Radicals
Exponents and Radicals Practice
Solving Radical Equations
Solving Radical Equations
Solving Radical Equations
Lecture-Radical Expressions
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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.

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