Math

Real numbers

All rational and irrational numbers

Natural Numbers

Also called the positive integers, are the counting numbers. N = {1,2,3...}

Simplifying radical expressions by rationalizing the denominator

a method used to eliminate radicals from the denominator of a fraction

Function

a rule that relates how the dependent variable is affected by the independant varialble. In the relation, no two different ordered pair can have the same x coordinate

PEMDAS

Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

Adding and Subtracting factions (same denominator)

1. Add or subtract the numerators. 2. If the answer is an improper form, reduce the fraction into a mixed number.

Converting improper fractions to mixed numbers

1. Divide the numerator by the denominator. 2. Write down the whole number answer. 3. Then write down any remainder above the denominator.

Adding and subtracting fractions (different denominator)

1. Find the Lowest Common Multiple (LCM) between the denominators. 2. Multiply the numerator and denominator of each fraction by a number so that they have the LCM as their new denominator. 3. Add or subtract the numerators and keep the denominator the same. 4: If the answer is an improper form, reduce the fraction into a mixed number.

Definition of higher-order roots

1. If x is a nonnegative real number, then ⁿ√x is a nonnegative nth root and has a property of (ⁿ√x)ⁿ = x. 2. if x is a negative real number, then a) (ⁿ√x)ⁿ if n is an odd integer b) (ⁿ√x)ⁿ is not a real number when n is an even inerger.

procedures for solving radical equations

1. Isolate the radical on one side. 2. Raise each side to the appropriate power to remove the radical. 3. Simplify, if possible. 4. If the equation still contains a radical, repeat steps 1-3. 5. collect all term on one side of the equation and set the equation to 0. 6. Solve the resulting equation. 7. Check all apparent solutions.

Converting mixed numbers to improper fractions

1. Multiply the whole number part by the fraction's denominator. 2. Add that to the numerator. 3. Then write the result on top of the denominator.

order of operations

1. always left to right 2. PEMDAS

Percent

1. means part of a hundred. 30% means 30 parts per 100 (%30 = 0.30) 2. to find the percent of a number, change the percent to a decimal and multiply the number by the decimal. %30 of 67; 0.30 • 67 = 20.1 3. to answer questions like " what percent is 36 out of 48?" we write the fraction of 36 out of 48 (36/48). Then change the fraction to a decimal (36/48 = 0.75). Then we write the decimal as a percent. (0.75 = %75)

Rules for adding, subtracting, multiplying, and dividing real numbers

1. to add two numbers with the same sign, add their absolute values. the sum takes the common sign. 2. to add two number with different signs, find the difference of their absolute values. The answer takes the sign of the number with the larger absolute value. 3. to substract b from a, add the inverse of b to a. thus a-b = a + (-b) 4. when multipying or dividing two real numbers with different signs, the answer is always negative. 5. when multiplying or dividing two real numbers with like signs, the answer is always positive.

improper fraction

A fraction whose numerator is larger than the denominator

Prime numbers

A natural number with only two factors, 1 and itself (2 is the only even prime number)

mixed number

A number made up of a whole number and a fraction

Rational Expression

A polynomial divided by a polynomial

Radical Expression

A radical expression is defined as any expression containing a radical (√) symbol.

Exponent

Or power is the number that tells how many times the base should be multiplied by itself

Dividing radical expressions

For all nonnegative real numbers a, all positive real numbers b, and positive integers n. ⁿ√a/ⁿ√b = ⁿ√(a/b)

Inverse Property

For any real number a there is a unique number -a such that if we add them we obtain the identity element - zero

Rational Numbers

Includes the integers and all quotients of integers; but division by zero is not allowed. It can be written as a terminating decimal (1/8 = 0.125) or as a repeating decimal (2/3 = 0.6666...)

Base

It is the number being multiplied to itself, like the number 2 in 2⁴.

Adding and subtracting radicals

Only like radicals can be added or subtracted. radicals are like when they have the same radicand and index.

Range

The complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain. The range is the resulting y-values we get after substituting all the possible x-values.

domain

The complete set of possible values of the independent variable. All possible x-values which will make the function "work", and will output real y-values.

Conjugates

The expressions a+√b and a-√b where a and b represent any algebraic term. each expression is the conjugate of the other

coefficient

The numerical factor of a term in a polynomial. Example: 14 is the coefficient in the term 14x

Commutative Property

The other in which numbers are added does not affect the sum. the other in which numbers are mutiplied does not afferct​ the product

Associative Property

The way in which numbers are grouped does not affect the sum (applies to multiplication)

Zero power

any number raised to the zero power equals 1

Relation

any set of ordered pairs

multiplication property of zero

anything multiplied by zero is equal to zero

Whole Numbers

are natural numbers plus zero (0). W = {0,1,2,3...}

Irrational Numbers

are numbers whose decimal forms are nonterminating and nonrepeating

Intergers

are whole numbers plus the negative of all natural numbers. I = {-3, -2, -1, 0, 1, 2, 3,}

division property of zero

division by zero is undefined

imaginary number

i = square root of (-1) and i square = (-1)

Product Rule of Exponents

if a is a real number and m and n are integers

Quotient rule of exponents

if a is a real number and m and n are integers

Identity Property

if we add a unique real number to any other real number, that number is not changed. Zero is the only unique number that has this property.

negative exponents

if x is any nonzero real number and n is an integer, x⁻ⁿ = 1/xⁿ

Rewriting roots as rational exponents

if x ≠ 0 and the power is negative then we can change the negative power to positive using the negative exponent rule x⁻ⁿ = 1/xⁿ

Solving radical equations by raising each side of the equation to a power

if y = x, then yⁿ = xⁿ, for all natural numbers.

Absolute Value

the distance of a number from zero

exponential notation

used to indicate repeated multiplication. 2³ = 2 • 2 • 2

Multiplying radical expressions

ⁿ√aⁿ√b = ⁿ√ab