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.
