GRE math concepts

Rules of exponents: (s^7)(t^7) =_

(st)^7

In any normal distribution, almost all data values (95%) fall within how many standard deviations from the mean?

*Almost all data values will fall within 2 standard deviations from the mean* This means that less than 5% will fall beyond 2 standard deviations from the mean

Negative exponents: (w^5)^-3 =_

1 / w^15 Other examples: x^-4 = 1/x^4 x^2 / x^-3 = (x^2)(x^3) = x^5 2x^-1 = 2/x (3x)^-2 = 1 / 9x^2

Because products of the form n(n-1)(n-2)...(3)(2)(1) occur frequently when counting objects, a special symbol n! called n factorial is used to denote this product.

1! = 1 2! = (2)(1) = 2 3! = (3)(2)(1) = 6 4! + (4)(3)(2)(1) = 24 * 0! = 1 ^So if there is one object being counted, there are 1! possible combinations, equalling one combination. If there are two objects being counted, there are 2! possible combinations, equalling 2 combinations, etc

What is a regular polygon?

A polygon in which all sides are congruent and all interior angles are congruent.

What are prime numbers?

A prime number is a whole number greater than 1, whose only two whole-number factors are 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. *An integer greater than 1 that is not a prime # is a composite number

What is the relationship between triangles, quadrilaterals & pentagons?

A quadrilateral can be divided into 2 triangles & a pentagon can be divided into 3 triangles. *If a polygon has n sides, it can be divided into (n-2) triangles.

What is an isosceles triangle?

A triangle with at least 2 congruent sides is an isosceles triangle. If a triangle has 2 congruent sides, then the angles opposite those 2 sides are congruent. The converse is also true (the sides opposite the congruent angles are equal in length).

Fractional exponents

An exponent of 1/2 is actually square root And an exponent of 1/3 is cube root An exponent of 1/4 is 4th root....etc...

What are congruent angles?

Angles that have equal measures (opposite angles have equal measures, thus opposite angles are congruent) 2 lines that intersect to form 4 congruent angles are called perpendicular lines

In a circle, any diameter is longer or shorter than any other chord that is not a diameter?

Any diameter is LONGER than any other chord that is not a diameter of the circle

Common circle equations

Area = pi r^2 Circumference = 2 pi r = pi diameter

What is the difference between quartiles and percentiles?

Besides the beginning, middle and end (or least, greatest and median of any list of data), the most common measures of position are quartiles and percentiles. They divide the data into roughly equal groups after the data have been ordered from the least to greatest values. There are 3 quartile numbers that divide the data into 4 roughly equal groups and there are 99 percentile numbers that divide the data into 100 roughly equal groups.

How to find the quartiles of a list of data

First divide the data into 2 groups (if it is a list of 16 numbers, each group will have 8 numbers in it). The second quartile (Q2) will be the median number. To find the other quartiles, take each of the smaller groups and find its median.

How to find the area of a parallelogram versus a trapezoid

For all parallelograms (including rectangles & squares), area equals base times height. For trapezoids, area equals half the product of the sum of the lengths of the 2 parallel sides and the corresponding height: (a+b /2)h

The probability of an event is a number from 0 to 1 that indicates the likelihood that the event occurs when the experiment is performed. The greater the number, the more likely the event. How is this calculated?

For any event E, the probability that E occurs, denoted by P(E), is defined by the ratio P(E) = # of outcomes in the event E / # of possible outcomes in the experiment In the case of rolling a 6-sided die, the 6 outcomes are equally likely. The probability of rolling a 4 is 1/6 and probability of rolling an odd number is 3/6=1/2. If an event is certain to occur P(E) =1 If an event is certain not to occur P(E)=0 If an event is possible but not certain to occur 0<P(E)<1 The probability that an event will not occur is 1 - P(E)

Basics of functions

For any function, each input (x) gives exactly one output f(x). The domain of a function is the set of all permissible units (all permissible values for the variable x).

How to calculate standard deviation

For the five data 0, 7, 8, 10, and 10 the standard deviation can be computed as follows. Find the mean: 7 The squared differences of the mean are (7-0)^2, (7-7)^2, (7-8)^2, (7-10)^2, (7-10^2) or 49, 0, 1, 9, 9 The average of the five squared differences is 68/5=13.6 Positive square root of 13.6 is approx 3.7

What are hundreds, tens and units digits?

Hundreds, tens, and units digits - The three digit number 728 (seven hundred and twenty eight) can be written 7 × 100 + 2 × 10 + 8. The hundreds digit is therefore 7, the tens digit is 2 and the units digit is 8.

How to find the vertex of a parabola

If a is positive, the parabola opens upward and the vertex is at its lowest point. If a is negative, the parabola opens downward and the vertex is at its highest point. When the equation is rewritten in vertex form like this, the vertex is the point (h,k): y=A(x−h)^2​​ +k Rewrite the parabola equation to y - c = ax^2 + bx We can complete the square by taking half of the coefficient of our x term, squaring it, and adding it to both sides of the equation. Now we can rewrite the expression in parentheses as a squared term. Move the constant term to the right side of the equation. Now the equation is in vertex form.

What does it mean if 2 events are independent?

If the occurrence of either event does not affect the occurrence of the other. Thus if events A and B are independent, the probability of A or B occurring is equal to P(A or B)= P(A) + P(B) - P(A)P(B)

What are the ratios of lengths of the sides of the 2 special right triangles?

Isosceles right triangle: 1 to 1 to sqrt of 2 30 - 60 - 90 triangle (half of an equilateral triangle): 1 to sqrt of 3 to 2

What is sample standard deviation?

It is computed by dividing the sum of the squared differences by n-1 instead of n

How to find the area of a parallelogram

Multiply the length of the base by the height corresponding to the base

How to find the sums of the interior angles of a polygon

Since the sum of the measures of the interior angles of a triangle is 180 degrees, it follows that the sum of the measures of the interior angles of an n-sided polygon is (n-2)(180 degrees). Ex: The sum of the interior angles of a quadrilateral is (4-2)(180) = (2)(180) = 360 degrees

How to find the y-intercept of a parabola

Since the y-intercept marks the point where x =0, all that you have to do is substitute 0 in for x in the parabola's equation.

Multiplication principle

Suppose there are 2 choices to be made sequentially and that the second choice is independent of the first choice. Suppose there are k different possibilities for the first choice and m different possibilities for the second choice. The multiplication principle states that under those conditions, there are km different possibilities for each pair of choices.

How to find the x-intercepts of a parabola

The graph of a quadratic equation of the form y=ax^2 + bx + c is a parabola. The x-intercepts of a parabola are the solutions of the equation ax^2 + bx + c =0 (FACTOR!)

Center coordinates and radius of a circle

The graph of an equation of the form (x-a)^2 + (y-b)^2 = r^2 is a circle with its center at the point (a,b) and with radius r

How to find lengths of sides of a right triangle

The hypotenuse will always be longer than the sides of a right triangle

What are integers?

The numbers 1,2,3 and so on together with their negatives

What is standardization?

The process of subtracting the mean from each value and then dividing the result by the standard deviation. *In any group of data, most of the data are within 3 standard deviations above or below the mean*

How to find the length of an arc of a circle (an arc is the part of the circle containing any 2 points on the circle and all the points between them)

The ratio of the length of the arc to the circumference is equal to the ratio of the degree measure of the arc to 360 degrees

How to find the length of the 3rd side of a triangle, if given the lengths of the 2 other sides

The sum of the lengths of any 2 sides of a triangle must be greater than the 3rd side

An antiques dealer bought c antique chairs for a total of x dollars. The dealer sold each chair for y dollars. a) Write an algebraic expression for the profit (P) earned from buying & selling the chairs. b) Write an algebraic expression for the profit per chair.

a) P = cy -x b) Profit per chair = P/c = cy-x / c = y - x/c

(x^10)(y^-1) / (x^-5)(y^5) =_

x^15 / y^6

How to solve systems of equations i.e: x+y=24 x-y=18

Choose one equation and isolate one variable; this equation will be considered the first equation. To isolate the y variable we subtract x from both sides. 2. Substitute the solution from step 1 into the second equation and solve for the variable in the equation.

Suppose that n objects are to be ordered from 1st to nth and we want to count the number of ways the objects can be ordered. What is each order called and how could we do this?

Each order is called a permutation. There are n choices for the first object, n-1 choices for the second object, n-2 choices for the third object, and so on, until there is only 1 choice for the nth object. Thus applying the multiplication principle, the number of ways to order the n objects is equal to the product n(n-1)(n-2)....(3)(2)(1) The product above is the number of permutations of n objects.

Basic properties of triangles

Each triangle has 3 sides & 3 interior angles. The interior angles add up to 180 degrees. The length of each side must be less than the sum of the lengths of the other 2 sides. Special triangles: Equilateral triangle (3 congruent sides & 3 congruent interior angles) Isosceles triangle (a triangle with at least 2 congruent sides) Right triangle 2 triangles that have the same shape & size are congruent triangles. 2 triangles that have the same shape but not necessarily the same size are similar triangles.

What is an equilateral triangle?

One which is 30 degrees - 60 degrees - 90 degrees

Special quadrilaterals

Parallelogram (both pairs of opposite sides are parallel; opposite sides are congruent and opposite angles are congruent) Trapezoid (2 opposite sides are parallel)

What is prime factorization?

Prime factorization or integer factorization of a number is the determination of the set of prime numbers which multiply together to give the original integer. The prime factorization of a square # must contain only even exponents.

Measures of dispersion to indicate the degree of "spread" of data

Range (difference between the greatest number and least number; G - L) Interquartile range (difference between the third quartile and first quartile; Q3 - Q1) Standard deviation (calculate the mean of the n values; find the difference between the mean and each n value; square each of the differences; find average of the n squared differences; take nonnegative square root of the average squared difference

Volume & surface area of a rectangular solid

Volume = length x width x height Surface area = 2(lengthxwidth + lengthxheight + widthxheight) OR the sum of the areas of the 6 faces

Volume and surface area of a right circular cylinder

Volume = pi x r^2 x height Area = 2(pi r ^2) + 2(pi r h)

Rules for solving inequalities

These are things we can do without affecting the direction of the inequality: Add (or subtract) a number from both sides. Multiply (or divide) both sides by a positive number. Simplify a side. But these things will change the direction of the inequality ("<" becomes ">" for example): Multiply (or divide) both sides by a negative number Swapping left and right hand sides

What does it mean if 2 events are mutually exclusive?

They cannot occur at the same time. Thus if events A and B are mutually exclusive, the probability of A or B occurring is equal to P(A or B)= P(A) + P(B)

How to calculate the weighted mean

When several values are repeated in a list, it is helpful to think of the mean of the #s as a weighted mean of only those values in the list that are different. Ex: 2,4,4,5,7.7.7.7.7.7,8,8,9,9,9,9 There are only six different values in the list: 2,4,5,7,8 and 9. The mean of the #s in the list can be computed as: 1(2) + 2(4) + 1(5) + 6(7) + 2(8) + 4(9) / 1 + 2 + 1 + 6 + 2 + 4 = 109/16 = 6.8125

How to find x & y intercepts of a straight line

To find the x-intercept of a given linear equation, plug in 0 for 'y' and solve for 'x'. To find the y-intercept, plug 0 in for 'x' and solve for 'y'. In this tutorial, you'll see how to find the x-intercept and the y-intercept for a given linear equation. Check it out!

What are similar triangles?

Triangles that are the same shape but not necessarily the same size. 2 triangles are similar if their vertices can be matched up so that the corresponding angles are congruent. When we say triangles ABC and DEF are similar, it is assumed that angles A and D are congruent, B and E are congruent, and C and F are congruent. In this case, AB / DE = BC / EF = AC / DF

How to find the number of ways to select and order k objects out of n objects (the number of ways to select and order k objects from a set of n objects)

n! / (n-k)! Ex: How many different 5-digit positive integers can be formed using the digits 1,2,3,4,5,6 and 7 if non of the digits can occur more than once in the integer? 7! / (7-5)! = (7)(6)(5)(4)(3)(2)(1) / 2! = 2520

How to find the number of combinations of n objects taken k at a time

n! / k!(n-k)! Ex: How many ways are there to select a 3-person committee from a group of 9 students? 9! / 3!(9-3)! = (9)(8)(7) / (3)(2)(1)= 84

Quadratic equation

x = -b +- sqrt(b^2 -4ac) / 2a