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College Algebra . Introduction P1 The Real Number System P2 Integer and Rational Number Exponents P3 Polynomials. Introduction . Welcome! Addendum Quarter Project Wikispaces website : http://msbmoorheadmath.wikispaces.com / Questions?!. P1 The Real Number System.
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College Algebra Introduction P1 The Real Number System P2 Integer and Rational Number Exponents P3 Polynomials
Introduction • Welcome! • Addendum • Quarter Project • Wikispaces website: • http://msbmoorheadmath.wikispaces.com/ • Questions?!
P1 The Real Number System • Bonus opportunity for the beginning of P1 on the Wikispaces site! • Number system • Prime/Composite Numbers • Absolute Value • Exponential Notation • Order of Operations
P1 - Evaluate • To evaluate an expression, replace the variables by their given values and then use the Order of Operations. • when x = 2 and y = -3
P1 - Evaluate You try! Evaluate when x = 3, y = -2 and z = -4
P1 – Properties of Addition Closure a + b is a unique number Commutative a + b = b + a Associative (a + b) + c = a + (b + c) Identity a + 0 = 0 + a = a Inverse a + (-a) = (-a) + a = 0
P1 – Properties of Multiplication Closure ab is a unique number Commutative ab = ba Associative (ab)c = a(bc) Identity a·1 = 1·a = a Inverse
P1 – Property Identification Which property do each of the following use?
P1 – Property Identification Which property do each of the following use?
P1 – Property Identification Which property do each of the following use?
P1 – Property Identification We use properties to simplify: First we will use the Commutative Property Then we will use the Associative Property
P1 – Property Identification We use properties to simplify: We will use the Distributive Property
P1 – Property Identification Use properties to simplify:
P1 – Property Identification Use properties to simplify:
P1 – Properties of Equality Reflexive a = a Symmetric If a = b, then b = a Transitive If a = b and b = c, then a =c Substitutional If a = b, then a may be replaced by b in any expression that involves a.
P1 – Properties of Equality Identify which property of equality each equation has:
P1 – Properties of Equality Identify which property of equality each equation has:
P1 Time for a break!
P2 – Integer Exponents Remember…. . Multiplied n times. So, , Be careful….
P2 – Integer Exponents If b ≠ 0 and n is a natural number, then and Examples:
P2 – Integer Exponents Examples: You try:
P2 – Integer Exponents You try:
P2 – Properties of Exponents Product Quotient where b ≠ 0 Power where b ≠ 0
P2 – Properties of Exponents Simplify: Simplify:
P2 – Properties of Exponents Simplify: Simplify:
P2 – Properties of Exponents Simplify:
P2 – Properties of Exponents You Try:
P2 – Properties of Exponents You Try:
P2 – Scientific Notation A number written in Scientific Notation has the form: Where n is an integer and • For numbers greater than 10 move the decimal to the right of the first digit, n will be the number of places the decimal place was moved 7, 430, 000
P2 – Scientific Notation For numbers less than 10 move the decimal to the right of the first non-zero digit, n will be negative, and its absolute value will equal the number of places the decimal place was moved 0.00000078
P2 – Scientific Notation Divide:
P2 – Rational Exponents and Radicals If n is an even positive integer and b ≥ 0, then is the nonnegative real number such that If n is an odd positive integer, then is the real number such that because
P2 – Rational Exponents and Radicals Examples: because because because because
P2 – Rational Exponents and Radicals Examples: However… ( is not a real number because If n is an even positive integer and b < 0, then is a complex number….we will get to that later…
P2 – Rational Exponents and Radicals For all positive integers m and n such that m/n is in simplest form, and fro all real numbers b for which is a real number. Example:
P2 – Rational Exponents and Radicals Example:
P2 – Rational Exponents and Radicals Simplify:
P2 – Rational Exponents and Radicals You Try:
P2 –Radicals Radicals are expressed by , are also used to denote roots. The number b is the radicand and the positive integern is the index of the radical. If n is a positive integer and b is a real number such that is a real number, then If the index equals 2, then the radical also known as the principle square root of b.
P2 –Radicals For all positive integers n, all integers m and all real numbers b such that is a real number, This helps us switch between exponential form and radical expressions
P2 –Radicals We can evaluate… Try on our calculator! http://web2.0calc.com/
P2 –Radicals If n is an even natural number and b is a real number, then If n is an odd natural number and b is a real number, then
P2 –Radical Properties If n and m are natural numbers and a and b are positive real numbers, then… Product Quotient Index
P2 –Radical Properties If n and m are natural numbers and a and b are positive real numbers, then… Product Quotient Index
P2 –Radicals How do we know if our expression is in simplest form? • The radicand contains only powers less than the index. • The index of the radical is as small as possible. • The denominator has been rationalized. Such that no radicals occur in the denominator. • No fractions occur under the radical sign.
P2 –Radicals Simplify:
P2 –Radicals Simplify:
P2 –Radicals Like radicals have the same radicand and the same index…
P2 –Radicals Simplify:
