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Math agc1 how to do problems and definitions
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I put this together to study for my test. I tried to provide more than just definitions. I hope this helps others as well as myself.
Terms in this set (87)
Whole Numbers
The numbers 0, 1, 2, 3, 4, 5, ... (and so on) (no fractions, decimals, or negatives)
Natural Numbers
Whole Numbers, but without the zero. Because you can't "count" zero. So they are 1, 2, 3, 4, 5, ...
Integers
Like whole numbers, but they also include negative numbers ... but still no fractions allowed! (Whole numbers and their opposite.)
Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }
Real Numbers
All numbers
Rational Number
A number that can be expressed as an integer or fraction.
The numerator and the denominator of the fraction are both integers.
When the fraction is divided out, it becomes a terminating or repeating decimal. (The repeating decimal portion may be one number or a billion numbers.)
Irrational Number
A real number that cannot be written as a simple fraction - the decimal goes on forever without repeating.
The square root of a whole number that is not a perfect square.
Example: Pi is an irrational number.
Prime Numbers
A Prime Number can be divided evenly only by 1 or itself.
It must be a whole number greater than 1.
Composite Numbers
Are made up of Prime Numbers multiplied together. A whole number that can be divided evenly by numbers other than 1 or itself
Factors of a Number
The numbers you multiply together to get another number.
Exponential Notation of a Factored Form
A number that represents how many times another number is being multiplied. 5x5x5 = 5 to the third power
Prime Factorization
Finding which prime numbers multiply together make the original number.
Greatest Common Factor (GCF)
The highest number that divides exactly into all the given numbers.
If you find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.
Least Common Multiple
The smallest common multiple (other than zero) of a list of numbers.
Exponent
How many times to use the number in a multiplication.
Shorthand for repeated multiplication. The rules for performing operations involving exponents allow you to change multiplication and division expressions with the same base to something simpler.
Exponents: Product with the same base:
When multiplying like bases, keep the base the same and add the exponents.
Exponents: Quotient with the same base:
When dividing like bases, keep the base the same and subtract the denominator exponent from the numerator exponent.
Exponents: Quotients to a power:
When raising a fraction to a power, distribute the power to each factor in the numerator and denominator of the fraction.
Exponents: Negative power:
Negative exponents signify division. In particular, find the reciprocal of the base.
When a base is raised to a negative power, reciprocate (find the reciprocal of) the base, keep the exponent with the original base, and drop the negative.
Exponents: Power to a power:
When raising a base with a power to another power, keep the base the same and multiply the exponents.
When raising a product to a power, multiply the power to each factor.
Exponents: Zero Power:
Anything raised to the zero power is one.
Exponents: Quotient with a Negative Power:
Negative exponents signify division - find the reciprocal of the base. When a denominator is raised to a negative power, move the factor to the numerator, keep the exponent but drop the negative.
1/5⁻³ (with five to the -3rd power) = 5³
Exponential Notation
Way of showing how many times a number or variable is used as a factor. Exponent expressed out: 5x5x5=125
Absolute Value
How far a number is from zero (no negatives).
Remove any negative sign in front of a number, and think of all numbers as positive (or zero).
Square Root
Multiply the number by itself
Addition Property of Equality
If A,B,C are real numbers, then: A=B and A+C = B+C
(you can add the same number to each side without changing the equation)
Solve x-16=7
1. Isolate "x" by adding 16 to both sides
2. x=23
Solving Two Step Equations
Always isolate the x
2x + 8 = 14
1. Subtract 8 from both sides (2x=6)
2. Drill down further to isolate the x: Divide both sides by 2
Answer: x=3
Removing Parenthesis Rule Presided by Minus or - Sign
All items in parenthesis change to the opposite sign.
Multiplication Property of Equality
If A,B, and C (C does not equal 0) represent real numbers, then:
A=B and AC = BC
(You can multiply each side of the equation by the same nonzero number without changing the solution.)
Removing Parenthesis Rule Presided by Plus or + Sign
All items in parenthesis keep the same sign
Order of Operations
PEMDAS
P
Parentheses first
E
Exponents (ie Powers and Square Roots, etc.)
MD
Multiplication and Division (left-to-right)
AS
Addition and Subtraction (left-to-right)
Circumference
The perimeter -- the distance around the outer edge.
= 2 pi radius
pi x diameter
Convert percent to fraction
Step 1: Write down the percent divided by 100.
Step 2: If the percent is not a whole number, then multiply both top and bottom by 10 for every number after the decimal point. (For example, if there is one number after the decimal, then use 10, if there are two then use 100, etc.)
Step 3: Simplify (or reduce) the fraction
Convert Percent to Decimal
Divide by 100, and remove the "%" sign.
Fundamental Counting Principle
The total number of different ways an event can occur.
If there are A ways for one activity to occur, and B ways for a second activity to occur, then there are a • b ways for both to occur.
(Simply multiply A and B)
Fundamental Counting Principle (more than two activities)
Multiply the number of choices for step 1 by the choices for step 2, step 3, etc...
Example:
A coin is tossed 5 times: (2 heads/tails choices for each toss) Multiple 2 by itself 5 times.
2x2x2x2x2=32 (32 different combinations)
Empirical Probability
An "estimate" that the event will happen based on how often the event occurs.
Example: Students choose their favorite animal. There are 85 animals and each student can only choose one.
10 Cats/ 15 Birds/ 35 Dogs/ 8 Rabbits/ 5 Lizards/ 12 Others
What is the probability that a student's favorite animal is a dog? Answer: 35 out of the 85 students chose a dog. The probability is (be sure to reduce):
35
__
85
Theoretical Probability
The number of ways that the event can occur, divided by the total number of outcomes. The probability of events that come from a sample space of known equally likely outcomes.
Example: Find the probability of rolling a six on a fair die.
(You could roll a 1, 2, 3, 4, 5, or 6 on one roll: six choices)
1
__
6
Example: Roll a fair die and getting an odd number. There are three of six odd numbers. Reduce to 1/2.
Perimeter
The distance around the outside of a figure. Add together the lengths of all of the sides of the figure.
Linear Equations
Undoing operations that are being done to the variable. The task is always to isolate the variable -- get the variable ALONE on one side of the equal sign.
Keep the equation "balanced" by making the same changes to BOTH sides of the equal sign.
Linear Inequalities
The same as solving linear equations...with one very important exception... when you multiply or divide an inequality by a negative value, it changes the direction of the inequality.
Positive Slope
Slant "up hill" (as viewed from left to right).
Negative Slope
Slant "down hill" (as viewed from left to right).
Front End Rounding
Rounding to the highest possible place so that all digits become zero except first one. Do not round up a number less than 10.
Improper Fraction
A fraction whose numerator is larger than the denominator
Proper Fraction
A fraction with a numerator smaller than the denominator
Mixed Number
A whole number and a fraction
Change Mixed Number to Improper Fraction
Multiply the whole number by the denominator, adding it to the numerator and placing all that over the denominator again. Also called Fractional Notation.
Multiply Fractions
Does not require a common denominator. Multiply the top (numerator) by the bottom (denominator) together and reduce. If asked to "simplify": if the numerator is larger than the denominator change to a "Mixed Number"
Dividing Fractions
Invert the 2nd fraction and multiply. Multiply the first fraction by the reciprocal of the second.
Reciprocal
One of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2; a fraction that has been flipped. The reciprocal of 3/4 is 4/3.
Quotient
Answer to a division problem.
Adding or Subtracting Fractions
Both fractions must have the same denominator. If the denominators are different rewrite them as equivalent with common denominators.
Opposites
Equal distance, but different directions from 0 on the number line. Also called additive inverse. -3 opposite is 3. -2/9 opposite is 2/9
Distributive Property
a(b + c) = ab + ac
Distribute multiplication to terms within the parenthesis and the parenthesis are removed.
If a,b, and c are real numbers,
then a(b+c) = ab + ac
Example: 3(x-4) = 3× x + 3 × (-4)
= 3x - 12
Simplifying Expressions
Reducing an expression by combining like terms and using order of operations.
Step #1 Simplifying Expressions: Identify and Combine Like terms:
Identify: Like terms have the same configuration of variables, raised to the same powers. They must have the same variable or variables, or none at all, and each variable must be raised to the same power, or no power at all.
Example: 1 + 2x - 3 + 4x.
Combine: Add terms together (or subtract in the case of negative terms) to reduce each set of terms with the same variables and exponents to one singular term.
Example: 2x and 4x and 1 and -3 are like terms
1 + -3 = -2
2x + 4x = 6x
1 + -3 = -2
Step #2 Simplifying Expressions: Simplify the Expression
Construct an expression from your new, smaller set of terms. Get a simpler expression that has one term for each different set of variables and exponents in the original expression. This new expression is equal to the first.
New expression is 6x - 2. This simplified expression is equal to the original (1 + 2x - 3 + 4x), but is shorter and easier to manage. It's also easier to factor.
Step #3 Simplifying Expressions: Order of operations
Use the acronym PEMDAS to remember the order of operations.
-Parentheses
-Exponents
-Multiplication
-Division
-Addition
-Subtraction
Ratio
Comparison of two quantities expressed as a quotient. Ratio of 7 to 5 = 7/5
Rate
A ratio that compares two quantities measured in different units.
Unit rate
The denominator is 1 when dividing the numerator by the denominator.
Algebraic Expression
An expression containing at least one variable (a letter).
The product of 3 and 3 more than a number: 3(x+3)
The difference between 3 times a number and 1: 3(x)-1
Solution Set
Set of all possible solutions of an equation.
Solve an equation to get a solution set:
/3x/ = 17.4 (Two possibilities 17.4 and -17.4)
Divide 3 by both possibilities
Solution set: {-5.8,5.8}
/2x-7/=18 (Two possibilities 18 and -18)
Isolate the "x" by adding 7 to both sides: 2x=25 and 2x= -11
Solution set:{25/2,-11/2}
Ordered Pair
A pair of numbers that can be used to locate a point on a coordinate plane.
y coordinate
the second coordinate in an ordered pair; tells you how far to go up or down
X Axis
the horizontal line on a graph
Y Axis
the vertical line on a graph
Table of Values
a table that contains several ordered pairs that make an equation in two variables true.
Similar Triangle
triangles that are the same shape, but different sizes
Congruent Triangle
Triangles where all three sides and all three angles are equal
Fraction
Another way to express division. Top number is the numerator and bottom is denominator.
Reducing fractions
Divide both the numerator and denominator by common factors:
Mixed numbers
A whole number and a fraction together.
Converting mixed number to improper fraction
Properties of addition
Commutative, Associative, and Additive identity
Property of zero
The Identity Property of Zero states that the sum of zero and any number is that number.
Inverse property of addition
add a numbers additive inverse to that number to cancel it out/equal zero.
Commutative property of addition
The property that says that two or more numbers can be added or multiplied in any order without changing the result.
If a and b are real numbers, then a+b=b+a.
Associative property of addition
The property that says that the sum or product of three or more numbers is always the same, no matter how you group them.
Properties of multiplication
commutative, associative, identity, property of zero, property of opposite
Property of One
a property that states that any number multiplied by 1 will equal that number
Zero exponent rule
Any base raised to the zero power is equal to 1
Commutative property of multiplication.
If a and b are real numbers,
then ab = ba
Example: (-4)(6)=-24 and
(6)(-4)=-24
Associative property of multiplication.
If a,b,and c are real numbers,
then (a+b)+c=a+(b+c)
Example: (-8×-2)×3 = 16×3=48 and
-8×(-2×3)=-8×(-6)=48
Factoring to simplify fractions
Original example expression, 9x2 + 27x - 3, is the numerator of a larger fraction with 3 in the denominator. This fraction would look like this: (9x2 + 27x - 3) ÷ 3. We can use factoring to simplify this fraction.
Let's substitute the factored form of our original expression for the expression in the numerator: (3(3x2 + 9x - 1)) ÷ 3
Notice that now, both the numerator and the denominator share the coefficient 3. Dividing the numerator and denominator by 3, we get: (3x2 + 9x - 1) ÷ 1.
Since any fraction with "1" in the denominator is equal to the terms in the numerator, we can say that our original fraction can be simplified to 3x2 + 9x - 1.
Rules of factoring
Factoring: #1 Identify the greatest common factor:
Identify the greatest common factor in the expression. Factoring is a way to simplify expressions by removing factors that are common across all the terms in the expression. To start, find the greatest common factor that all of the terms in the expression share - in other words, the largest number by which all the terms in the expression are evenly divisible.
Example: 9x to the power of 2 + 27x - 3. Every term in this equation is divisible by 3. Since the terms aren't all evenly divisible by any larger number, we can say that 3 is our expression's greatest common factor.
Factoring #2: Dividing the terms by the GCF:
Divide every term in your equation by the greatest common factor. The resulting terms will all have smaller coefficients than in the original expression.
Factor out 9x ² + 27x -3 by its greatest common factor of 3. To do so divide each term by 3.
1. 9x² divided by 3 = 3x²
Equals: 27x divided by 3 = 9x
Equals: -3 divided by 3 = -1
Thus, our new expression is 3x2 + 9x - 1.
Factoring #3: Make new expression equal to the old one:
To make our new expression equal to the old one, we'll need to account for the fact that it has been divided by the greatest common factor. Enclose your new expression in parentheses and set the greatest common factor of the original equation as a coefficient for the expression in parentheses.
For the example expression, 3x² + 9x - 1:
Enclose the expression in parentheses and multiply by the greatest common factor of the original equation to get 3(3x (² + 9x - 1). This equation is equal to the original, 9x² + 27x - 3.
Use square factors to simplify radicals
Use square factors to simplify radicals:
Expressions under a square root sign are called radical expressions. Simplify by identifying square factors and perform the square root operation on these separately to remove them from under the square root sign.
Example: √(90)
If we think of the number 90 as the product of two of its factors, 9 and 10, we can take the square root of 9 to give the whole number 3 and remove this from the radical. In other words:
√(90)
√(9 × 10)
(√(9) × √(10))
3 × √(10)
3√(10)
Multiplying and Dividing Exponential Terms
Add exponents when multiplying two exponential terms; subtract when dividing. This concept can also be used to simplify variable expressions.
Example: 6x³ × 8x⁴ + (x¹⁷ - 15)
(6 × 8)³ + 4 + (x ¹⁷ - 15)
48x7 + x2
radical expression
an algebraic expression that includes a root. The root may be a square root, a cube root, or any other power. Simplifying a radical expression can help you solve an equation. Simplifying radical expressions involves removing the root when possible, or reducing the radicands, the numbers inside the radical symbol, as much as you can.
Simplify any radical expressions that are perfect squares:
A perfect square is the product of any number that is multiplied by itself, such as 81, which is the product of 9 x 9. To simplify a radical expression that is a perfect square, simply remove the radical sign and write the number that is the square root of the perfect square.
For example, 121 is a perfect square because 11× 11 is 121. You can simply remove the radical sign and write 11 as your answer.
Simplify any radical expressions that are perfect cubes:
A perfect cube is the product of any number that is multiplied by itself twice, such as 27, which is the product of 3× 3 × 3. To simplify a radical expression when a perfect cube is under the cube root sign, simply remove the radical sign and write the number that is the cube root of the perfect cube.
For example, 512 is a perfect cube because it is the product of 8 × 8 × 8. Therefore, the cube root of the perfect cube 512 is simply 8.
Break down an imperfect radical expression into its multiples:
STEP 1: The multiples are the numbers that multiply to create a number -- for example, 5 and 4 are two multiples of the number 20. To break down an imperfect radical expression by its multiples, write down all of the multiples of that number (or as many as you can think of, if it's a large number) until you find one that is a perfect square.
For example, all the multiples of the number 45: 1, 3, 5, 9, 15, and 45. 9 is a multiple of 45 that is also a perfect square. 9 × 5 = 45.
STEP 2: Remove any multiples that are a perfect square out of the radical sign: 9 is a perfect square because it is the product of 3 × 3. Take the 9 out of the radical sign and place a 3 in front of it, leaving 5 under the radical sign.
Find a perfect square in the variable
The square root of a to the second power would be a. The square root of a to the third power is broken down into the square root of a squared times a -- this is because you add exponents when you multiply variables, so that a squared times a reverts back to a cubed. Therefore, the perfect square in the expression a cubed is a squared.
Simplify a radical expression with variables and numerals that is a perfect square:
EXAMPLE: √36a²
take apart the expression by first looking for perfect squares in the numbers, and then looking for perfect squares in the variables. Then, remove the radical sign and let their square roots remain. the square root of 36 x a squared.
1. 36 is a perfect square because 6 × 6 = 36.
2. a squared is a perfect square because a times a is a squared.
Now that you've broken down these numbers and variables into their square roots, simply remove the radical sign and leave the square roots. The square root of 36 ×∧ a 6a.
Simplify a radical expression with variables and numerals that is not a perfect square:
√ 50a³
break down the expression into numerals and variables, and search for perfect squares within the multiples of both. Then, pull any perfect squares out of the radical expression.
-Break down 50 to find any multiples that are a perfect square. 25 × 2 = 50 and 25 is a perfect square because 5 × 5 = 25. To simplify root 50, you can pull a 5 out of the radical sign and leave 2 within it.
-Break down "a" to the third power to find any multiples that are a perfect square. a to the third power is really a squared times a, and a squared is a perfect square. You can pull one a out of the radical sign and leave one a inside the sign. Therefore, root a cubed is really a root a.
-Put it all together. Just place everything you took out of the radical sign out of the radical sign and keep everything that you kept in the sign in the sign. Combine 5 root 2 and a root a to create 5 x a root 2 x a
Polynomial
The sum or difference of one or more monomials. A polynomial with two terms is a binomial. With three terms is a trinomial The degree of a polynomial is the degree of the highest monomial term. A polynomial can have constants, variables and exponents, but never division by a variable. Can have just one constant. In graphing the lines are smooth and continuous. The standard form for writing a polynomial is to put the terms with the highest degree first
Adding/subtracting Polynomials
Step 1: Remove all parentheses. Write the problem vertically rather than horizontally because it makes the next step must easier. When adding, distribute the positive (or addition) sign, which does not change any of the signs. When subtracting, distribute the negative (or subtraction) sign, which changes each sign after the subtraction sign.
Step 2: Combine like terms. This step is much easier if things are written vertically because like terms are written above one another. Remember that to combine like terms the variable and the power of each variable must be exactly the same.
Distance formula
D=rate*time
Perimeter of a geometric figure
The distance around or the sum of the lengths of its sides:
Rectangle: 2 times the length plus 2 times the width.
Square: 4 times the length of the side.
Area: Always expressed in square units since its two-dimensional. Area of a rectangle: A = L*W
Function
1. A relationship from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. 2. The action or actions that an item is designed to perform.
Vertical line test
If any vertical line passes through more than one point of the graph, then the relation is a function
Increasing function
A graph that has a positive slope and shows growth
Decreasing function
A graph that has a negative slope and shows decay
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