# ALEKS Math Answer Key 2023 [FREE ACCESS]

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Then check out below questions and answers…

## All ALEKS Math Answers [K-12 to High ED]

Yes, we will be discussing all ALEKS Math topics such as Basic Math, Math placement test Q&A, and all previously asked questions in ALEKS for the grades K-12 to High ED.

Q. Improper fraction
Ans: A fraction whose numerator is larger than the denominator.

Q. Expanded form
Ans: The expanded form of a number shows the sum of its place values.

Q. Commutative Property
Ans: The order in which numbers are added or multiplied does not change the sum or product: a + b = b + a

Ans: If you add zero to a number, the sum is the same as that number: a + 0 = a

Ans: Changing the grouping of three or more addends does not change the sum: (a + b) + c = a + (b + c)

Q. GCF
Ans: Greatest Common Factor: The greatest number that is a factor of two or more given numbers.

Q. Factor
Ans: A whole number is a factor of another when it divides that number exactly

Q. Simplest form of a fraction
Ans: A fraction is in simplest form when the numerator and denominator have no common factors other than 1.

Q. Reciprocal
Ans: The multiplicative inverse of a number. A flipped fraction pair: 1/4 x 4/1 = 1

Q. Multiplicative inverse
Ans: A reciprocal of a number that is multiplied by that number resulting in a product of 1: A flipped fraction pair: 1/4 x 4/1 = 1

Q. Parallel
Ans: Two lines are parallel if they never cross.

Q. Perpendicular
Ans: Two lines are perpendicular if they cross at a right angle.

Q. Area
Ans: The area of a flat figure is the number of square units needed to cover the figure: Length x Width

Q. Perimeter
Ans: The distance around a figure.

Q. Evaluate
Ans: find the value of

Q. LCM
Ans: Least Common Multiple: The smallest number is a multiple of two or more given numbers.

Q. Mixed number
Ans: A number made up of a whole number and a fraction.

Q. Whole number
Ans: All positive, non-decimal, non-fractional integers.

Q. Integer
Ans: All whole numbers (both positive and negative) and zero.

Q. Proper fraction
Ans: A fraction with a numerator smaller than the denominator.

Q. Signed numbers
Ans: Numbers that are either positive or negative.

Q. Algebraic expression
Ans: An expression or formula with numbers, variables, and operations.

Q. Product
Ans: The answer to a multiplication problem

Q. Sum

Q. Quotient
Ans: The result of division

Q. Dividend
Ans: The number that is being divided

Q. Divisor
Ans: The number by which another number is divided.

Q. Remainder
Ans: The amount left over when one number is divided by another.

Q. Difference
Ans: The result of subtraction

Q. Minuend
Ans: A number from which another number is subtracted.

Q. Subtrahend
Ans: A number that is subtracted from another number

Ans: Any number is being added.

Q. Order of Operations
Ans: Parentheses, exponents, multiplications & divisions (from left to right), additions & subtractions (from left to right).

Q. PEMDAS
Ans: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

Q. Acute angle
Ans: Less than 90 degrees

Q. Obtuse angle
Ans: An angle between 90 and 180 degrees

Q. Right angle
Ans: 90-degree angle

Q. Identity Property of Multiplication
Ans: If you multiply a number by one, the product is the same as that number: a x 1 = a

Q. Associative Property of Multiplication
Ans: Changing the grouping of three or more factors does not change the product. (ab)c=a(bc)

Q. Zero Property of Multiplication
Ans: The product of any number and 0 is 0

Q. Commutative Property of Multiplication
Ans: Changing the order of the factors does not change the product: ab=ba

Q. Denominator
Ans: The bottom number in a fraction

Q. Nominator
Ans: The top number in a fraction.

Q. Mode
Ans: The value that occurs most frequently in a given data set.

Q. Range
Ans: It is the difference between the largest and smallest values in a data set. l – s = r

Q. Ratio
Ans: It is a comparison of two quantities, often expressed as a fraction or a quotient.

Q. Volume of a solid (a 3-dimensional object)
Ans: It is the amount of space it takes up. measured in cubic units.

Q. Cubic units of measure
Ans: An object’s length x width x height; measures an object’s volume or capacity.

Q. Compute
Ans: To determine by mathematics or calculation.

Q. Power (of a number)
Ans: Exponent; the number of times a base number is multiplied by itself.

Q. Exponent
Ans: How many times the base is used as a factor in the power of a number.

Q. 2 to the fourth power
Ans: 2^4

Q. The fourth power of 2
Ans: 2^4

Q. Powers of ten
Ans: 10^1, 10^2, 10^3, etc… the exponent in the power of 10 gives the number of zeroes.

Q. Power of 6
Ans: 6^x

Q. Fourth power of 10
Ans: 10’000

Q. Order of grouping symbols in algebra
Ans: {[()]}

Q. Supplement angle
Ans: One angle of a pair whose measures add up to 180

Q. Supplementary
Ans: Two angles that add up to 180 degrees

Q. Linear pair
Ans: A pair of adjacent angles whose non-common sides are opposite rays; any pair of linear angles are supplementary.

Q. Opposite rays
Ans: Two rays that have a common endpoint and form a line.

Q. Ray
Ans: A straight line extending from a point

Q. Linear Pair Property
Ans: If two angles form a linear pair, then they are supplementary

Q. Justification of the Linear Pair Property
Ans: Verifying the truth of the Linear Pair Property by adding the two angles. By the angle addition property m∠1m∠2m∠ABC, and by definition, m∠ABC180°. Thus, m∠1m∠2180°, and the angles must be supplementary.

Ans: Two or more adjacent angles can be added together to create a single larger angle

Q. Angle
Ans: A figure formed by two rays with a common endpoint

Q. Vertex angle
Ans: The angle is formed by the legs of an isosceles triangle.

Q. Isosceles triangle
Ans: A triangle with at least two congruent sides

Q. Congruent
Ans: Having the same size and shape

Q. Vertex of a polygon
Ans: A point at which two sides of a polygon meet. The plural of vertex is vertices.

Q. Polyhedron
Ans: A three-dimensional figure with faces that are polygons

Q. Faces
Ans: The flat surfaces of a 3-D figure

Q. Vertex of a polyhedron
Ans: A point where three or more edges meet

Q. Slant height
Ans: The height of each lateral face

Q. Lateral face
Ans: In a polyhedron, a face that is not a base.

Q. Polyhedron base
Ans: The face that is not a lateral face

Q. The altitude of a triangle
Ans: A perpendicular segment from a vertex to the line containing the opposite side

Q. The altitude of a polyhedron
Ans: A perpendicular segment from the base to the opposite vertex or face; height

Q. Complementary angles
Ans: Two angles whose sum is 90 degrees

Q. Complement angle
Ans: One angle of a pair whose sum is 90 degrees

Q. Vertical angles
Ans: A pair of opposite congruent angles formed by intersecting lines

Q. Distribute property
Ans: Multiplying a number by a sum or difference in parentheses.

Q. Prime number
Ans: A whole number that has exactly two factors, 1 and itself.

Q. LCD
Ans: Least Common Denominator is defined as the least common multiple of the denominators of two or more fractions.

Q. Sum of the angles in a triangle
Ans: 180°

Q. Squared Number
Ans: A number multiplied by itself or a number with an exponent of 2

Q. Square root
Ans: One of two equal factors of a number

Q. Scalene triangle
Ans: A triangle with no congruent sides; no sides (or no angles) have the same measure

Q. Equilateral triangle
Ans: A triangle with three congruent sides or all sides (or all angles) has the same measure.

Q. Divisibility rule for 3
Ans: A whole number is divisible by 3 when the sum of its digits is divisible by 3.

Q. Divisibility rule for 9
Ans: A whole number is divisible by 9 when the sum of its digits is divisible by 9.

Q. If a is positive, then a^n is?
Ans: Positive

Q. If a is negative and n is odd, then a^n is?
Ans: Negative

Q. If a is negative and n is even, then a^n is?
Ans: Positive

Q. Terminating decimal
Ans: A decimal whose digits end

Q. Repeating decimal
Ans: A decimal that repeats a digit or group of digits forever.

Q. 1 foot = ? inches
Ans: 12

Q. Prime factorization
Ans: Breaking down a composite number until all of the factors are prime

Q. Area of a circle
Ans: A=πr²

Q. Pythagorean Theorem
Ans: a²+b²=c²

Q. Hypothenuse
Ans: The side of a right triangle opposite the right angle

Q. Legs of a right triangle
Ans: The two sides that form the right angle

Q. Parallelogram
Ans: A quadrilateral with two pairs of parallel sides

Ans: A four-sided polygon

Q. Polygon
Ans: A closed plane figure made up of line segments

Q. Trapezoid
Ans: A quadrilateral with exactly one pair of parallel sides

Q. Area of a parallelogram
Ans: A=bh

Q. Area of a trapezoid
Ans: A=1/2h(b1+b2)

Q. Volume of a cylinder
Ans: V=πr²h

Q. Coefficient
Ans: A number multiplied by a variable in an algebraic expression.

Q. Interval notation
Ans: A notation for describing an interval on a number line. The interval’s endpoint(s) are given, and a parenthesis or bracket is used to indicate whether each endpoint is included in the interval.

Q. Standard notation
Ans: A number is written without exponents: an integer or a decimal fraction: 2.96 x 10^4= 29600

Q. Compound inequality
Ans: Two or more inequalities joined together by “and” or “or”

Q. Rational numbers are
Ans: Fractions

Q. Irrational numbers
Ans: Numbers that cannot be expressed in the form a/b, where a and b are integers and b =0.

Q. 1 oz = _ grams
Ans: 28.4 grams

Q. 1 liter = __ pints
Ans: 2.1 pints

Q. 1 mile = __ kilometers
Ans: 1.6 km

Q. 1 inch = _ cm
Ans: 2.54 cm

Q. Circumference defined as
Ans: Distance around the circle

Q. Circumference formula
Ans: C=2πr

Q. Area of a rectangle
Ans: A=lw

Q. Area of a triangle
Ans: A=1/2bh

Q. Area of a circle
Ans: A=πr²

Q. Area of a square
Ans: A=s²

Q. Volume of a Pyramid/Cone
Ans: v=1/3Bh

Q. Volume of a cylinder
Ans: V=Bh

Q. Volume of a sphere
Ans: 4/3πr³

Q. Integers are
Ans: Positive and negative whole numbers and zero

Q. Consecutive even integers
Ans: Even numbers that follow each other on a number line ie: −6,−4,−2,0… variable form: n, n+2, n+4, n+6

Q. Consecutive odd integers
Ans: Odd numbers that follow each other on a number line ie: −5,−3,−1,1… variable form: n, n+2, n+4

Q. Interest earned
Ans: Amount invested x rate = interest earned

Q. Distance
Ans: rate x time

Q. “And” in combined inequalities
Ans: Only looking for what they have in common

Q. “Or” in combined inequalities
Ans: Includes everything mentioned

Q. When would you change the direction of an inequality?
Ans: When you multiply or divide by a negative number.

Ans: (a+b)+c=a+(b+c)

Ans: Adding 0 to a number equals the original number

Ans: a+b=b+a

Ans: a+0=a

Q. Absolute value
Ans: The distance a number is from zero on a number line. ALWAYS POSITIVE, may represent units.

Q. Scalene triangle
Ans: A triangle with no congruent (equal) sides

Q. Median
Ans: the middle score in a distribution; half the scores are above it and half are below it

Q. Mean
Ans: the arithmetic average of a distribution, obtained by adding the scores and then dividing by the number of scores

Q. An ordered pair
Ans: Any pair of numbers for which order is important. Example: (x,y)

Q. Domain
Ans: The set of x-coordinates in a relation.

Q. Relation
Ans: A set of ordered pairs

Q. Range
Ans: The set of all y-values

Q. It is okay to have a zero in the _ of a fraction.
Ans: Numerator

Q. You cannot have a zero in the _ of a fraction.
Ans: Denominator

Q. What is a function?
Ans: Each x must be paired with only one y value

Q. Vertical line test
Ans: If any vertical line passes through no more than one point of the graph of a relation, then the relation is a function.

Q. A function can have the same y coordinate, but not the same x coordinate.
Ans: True

Q. Graphing Inequalities
Ans: A solid line means the inequality is ≥ or ≤. A dotted line means the inequality is > or <.

Q. How do you know which side to shade when graphing inequalities?
Ans: Create a test point (usually (0,0). If the resulting point is TRUE, Shade the side that includes the test point.

Q. 4 basic types of transformations
Ans: Reflection, rotation, translation, dilation

Q. For any flat line, the slope is always:
Ans: m=0

Q. For any vertical line, the slope is always:
Ans: Undefined, no slope

Q. Slope formula
Ans: m=y2-y1/x2-x1

Q. Slope-intercept form
Ans: y=mx+b, where m is the slope and b is the y-intercept of the line.

Q. Commutative Property
Ans: The property says that two or more numbers can be added or multiplied in any order without changing the result.

Q. Associative Property
Ans: The way in which numbers are grouped does not change their sum or product

Ans: If you add zero to a number, the sum is the same as that number.

Q. Velocity formula
Ans: v=d/t

Q. Factoring formula for the difference of squares
Ans: A^2 – B^2 = (a+b)(a-b)

Q. Rewrite the following without an exponent:
Ans:
(-4)^-3
A^-n=1/a^n
-1/64

Ans: -b±[√b²-4ac]/2a

Q. Slope
Ans: (y₂-y₁)/(x₂-x₁)

Q. Slope-Intercept
Ans: y=mx+b

Q. a³-b³
Ans: (a-b)(a²+ab+b²)

Q. a³+b³
Ans: (a+b)(a²-ab+b²)

Q. a²-b²
Ans: (a-b)(a+b)

Q. a²-2ab+b²
Ans: (a-b)²

Q. a²+2ab+b²
Ans: (a+b)²

Q. (a+b)(c+d)

Q. a(b+c)
Ans: ab+ac

Q. sine ratio
Ans: opposite ÷ hypotenuse

Q. cosine ratio

Q. tangent ratio

Q. Direct Variation
Ans: y=kx

Q. Inverse Variation
Ans: y=k/x

Q. Point-Slope form
Ans: y-y₁=m(x-x₁)

Q. Standard form
Ans: Ax + By=C, where A, B, and C are not decimals or fractions, where A and B are not both zero, and where A is not a negative

Q. Undefined
Ans: When there is a vertical line that has different y points, but the same x point

Q. Zero
Ans: When there is a horizontal line that has different x points, but the same y point

Q. Dividing by a negative number in an inequality
Ans: You must flip the sign

Q. Graphing < or > on a coordinate plane
Ans: Dotted line

Q. Graphing ≥ or ≤ on a coordinate plane
Ans: Solid line

Q. Graphing ≥ or > on a coordinate plane
Ans: Shade upwards or to the right

Q. Graphing ≤ or < on a coordinate plane
Ans: Shade downwards or to the left

Q. Infinitely many solutions
Ans: When the system of equations have the same slope and y-intercept

Q. One solution
Ans: When the system of equations have different slopes

Q. No solution
Ans: When the system of equations have the same slope but different y-intercepts

Q. Linear parent function
Ans: y=x or f(x)=x

Q. Elimination method
Ans: Solving systems by adding or subtracting equations to eliminate a variable

Q. Solution of the system of linear equations
Ans: Any ordered pair in a system that makes all the equations true

Q. Graphing method
Ans: Graphing the system of equations and finding the point at which they intersect

Q. Substitution method
Ans: Replacing one variable with an equivalent expression containing the other variable

Q. Absolute value equation
Ans: V-shaped graph that points upward or downward

Q. Translation
Ans: A shift of a graph horizontally, vertically, or both, results in a graph of the same shape and size, but in a different position.

Q. Area of a circle
Ans: Πr²

Q. Area of a square
Ans: s², where, s = length of a side

Q. Area of a triangle
Ans: ½(base x height) [or (base x height)÷2]

Q. Area of a trapezoid
Ans: ½(b₁ +b₂) x h [or (b₁ +b₂) x h÷2]

Q. Perimeter of a rectangle
Ans: 2Length + 2width [or (length + width) x 2]

Q. Perimeter of a square
Ans: 4s (where, s = length of a side)

Q. Area of rectangle, square, parallelogram
Ans: A=bh

Q. Area of a sector
Ans: x°/360 times (∏r²), where x is the degrees in the angle

Q. length of a sector
Ans: x°/360 times (2 pi r), where x is the degrees in the angle

Q. Circle
Ans: Is the set of points that are all the same distance (its radius) from a certain point( the center).

Ans: A segment connecting the center of a circle to any point on the circle

Q. Diameter
Ans: The distance across the circle through the center of the circle. The diameter is twice the radius.

Q. Chord
Ans: The distance from one point on the circle to another point on the circle.

Q. Sector
Ans: The part of a circle that looks like a piece of pie. A sector is bounded by 2 radii and an arc of the circle.

Q. Arc
Ans: Part of a circle connecting two points on the circle.

Q. Central Angle
Ans: An angle whose vertex is the center of the circle

### ALEKS Math Placement Test Answers

Below you can find answers for ALEKS Math Placement test questions that we collected from past 4 years to till now:

Q. Positive Number
Ans: Any number greater than zero

Q. Negative Number
Ans: Any number less than zero

Q. Even Number
Ans: Any integer divisible by 2

Q. Odd Number
Ans: Any integer not divisible by 2

Q. Sum
Ans: The result of two numbers being added

Q. Difference
Ans: The result of one number being subtracted by another

Q. Product
Ans: The result of two numbers being multiplied

Q. Quotient
Ans: The result of one number being divided by another

Q. Factor
Ans: A number that is multiplied to form the product

Q. Greatest Common Factor
Ans: The greatest whole number that is a factor of two or more numbers

Q. Multiples
Ans: The product of a number and a counting number

Q. Least Common Multiple
Ans: The least whole number that is a multiple of two or more numbers

Q. Prime
Ans: A number greater than one whose only whole number factors are one and itself

Q. Composite
Ans: A number greater than one with more than two whole-number factors

Q. Real
Ans: All positive and negative numbers and zero

Q. Rational
Ans: Any number that can be expressed as the quotient of two integers

Q. Irrational
Ans: Real numbers that are not rational

Q. How to find Percent Increase or Decrease
Ans: Difference/Original = Percent Increase or Decrease

Q. Average Problems
Ans: Average x Number = Total

Q. Simple Probability Problems
Ans: Outcomes giving desired result/Total possible outcomes

Q. Conditional Probability
Ans: The probability of an event, given another event, has already occurred. Ex: The probability that a playing card from a standard deck is a 5, given that it is red, is 2/26.

Q. Joint Probability
Ans: The probability of two events occurring at the same time. Ex: The probability that a playing card from a standard deck is red and a 5 is 2/52.

Q. graph ≥, ≤
Ans: solid line

Q. graph >, <
Ans: DOTTED line

Q. every, each, per (writing equations)
Ans: rate/ set y value equal to (multiply this value by the x variable)

Q. When writing an equation or working ANY inequality problem what does it help to do first?
Ans: Write out what’s happening

Q. What should you double-check for EVERY time
Ans: Writing problem, negative signs, uncarried negative signs, unflipped inequality signs, undotted lines, incorrect multiplication, fully simplified

Q. distance =
Ans: rate x time

Q. When solving a system of equations can you add the equations together if one of the variables will cancel out?
Ans: YES

Q. What should you always write out when doing your work?
Ans: Fraction conversions, simplification steps, line graphs (interval notation)

Q. What standard does an equation/ graph have to meet to be considered a function?
Ans: Each input can only have one output

Q. How do you translate a graph up or down?
Ans: The value outside of the parentheses (DONT change sign)

Q. How do you translate a graph side to side?
Ans: The value inside the parentheses (CHANGE SIGN)

Q. Translate -(x + 3)^3 – 2 from -x^3
Ans: left 3, down 2

Q. When translating do you need to solve the equation?
Ans: No, just look at the numbers inside (side to side change sign) and outside (up and down DONT change sign) the parentheses

Q. U represents
Ans: Numbers in EITHER set (Union)

Q. ⋂ represents
Ans: Numbers in BOTH sets (Intersection)

Q. What is the quadratic formula?
Ans: -b±√b^2-4ac/2a

Q. (x-6)(x+4) < 0
Ans: Where is the product of (x-6) and (x+4) less than 0?

Q. Should you graph two separate equations on the same number line when doing your work?
Ans: No

Q. When you take the square root of a variable, how many answers are there?
Ans: 2 answers (negative and positive)

Q. x^a/x^b
Ans: x^a-b

Q. x^a times x^b =
Ans: x^a+b

Q. (x^a)^b
Ans: x^ab

Q. What should you always physically mark on inequality graphs that are getting compared before attempting to solve the problem?
Ans: Where the graphs intersect

Q. BETC
Ans: Divide the top leading coefficient by the bottom leading coefficient (Bottom Equals Top Coefficient HORIZONTAL ASYMPTOTE)

Q. BOB0
Ans: The horizontal asymptote = 0 (Bigger On Bottom 0)

Q. BOTU
Ans: There is no horizontal asymptote, find slant asymptote (Bigger On Top Undefined)

Q. How do you find the slant asymptote?
Ans: Divide the ENTIRE numerator by the ENTIRE denominator, the slant asymptote is the quotient, ignore the remainder (might include division with variables raised to exponents)

Q. How do you find the vertical asymptote?
Ans: Set the denominator equal to 0

Q. sin
Ans: csc

Q. cos
Ans: sec

Q. tan
Ans: cot

Q. Can a graph have more than one local maximum or minimum?
Ans: yes

Q. volume
Ans: cubed

Q. area
Ans: squared

Q. perimeter/ length
Ans: no power

Q. How do you find the vertex of a parabola?
Ans: Take the derivative (slope) and set it equal to 0

Q. d/dx (x^a)
Ans: ax^a-1

Q. antiderivative (x^a)
Ans: (x^a+1)/a+1

Q. area of a circle
Ans: πr²

Q. Translate this equation to the right 3 and up 4:
Ans: f(x)=1/2x^2(1/2(x-3)^2)+4

Q. If a question asks for all real zeroes should you include imaginary numbers?
Ans: no

Q. In the functions (f + g)(x), (f-g)(1), (f・g)(x), (f/g)(2) etc. is the x or the coefficient in the second parentheses being multiplied by the solution of f and g?
Ans: No, x is the input for the combined functions (just like it would be in single functions like f(x))

Q. y=af(x)
Ans: multiply y coordinate by a

Q. y=f(ax)
Ans: divide x coordinate by a

Q. to find x intercept
Ans: set f(x) equal to 0

Q. to find y intercept
Ans: find f(0)

Q. What are inverse functions.
Ans: Functions that switch the input and output

Q. How do you find the inverse function of an equation?
Ans: Solve for x then swap the variables of the new solved equation

Q. What should you always do first
Ans: Cancel/ simplify/ factor

Q. When a problem says that one variable varies inversely with another variable what do you need to know/ find?
Ans: the constant k

Q. Direct variation
Ans: y=kx (ratio remains constant (y/x stays the same))

Q. Inverse variation
Ans: y=k/x (product remains constant (yx stays the same))

Q. Joint variation
Ans: y=kzx (y varies jointly with x and z (y/xz stays the same))

Q. When doing polynomial long division, which terms do you use

Q. What do you normally confuse the least common multiple with?
Ans: The greatest common factor (the least common multiple IS NOT the greatest common factor)

Q. How do you find the least common multiple?
Ans: Find the smallest whole number that both expressions divide evenly into. Then use the variables with the highest power

Q. What is the least common multiple of 25u^3v^4 and 10w^2u^5v^6 ?
Ans: 50u^5w^2v^6

Q. When performing operations on rational expressions, what does it help to circle?
Ans: Subtraction signs that apply to an entire expression (apply them at the very end)

Q. When should you find the Lowest Common Factor
Ans: ANY TIME you are adding or subtracting fractions, regardless of whether or not there are variables

Q. When performing operations on rational expressions, if there is a coefficient, do you need to multiply the coefficient by the lowest common factor as well?
Ans: YES

Q. Can you cancel out the denominator by multiplying both sides by its reciprocal regardless of whether or not there is a zero on the other side?
Ans: YES

Q. Can a solution make the denominator equal 0?
Ans: NO (ALWAYS CHECK FOR THIS BEFORE ANSWERING if a “solution” does this it is not a solution. it iS A FRAUD>~< )

Q. b√x^a
Ans: x^a/b

Q. x^a/b
Ans: b√x^a

Q. What is the vertex of this equation?
Ans: -3(x-5)+2 (the parentheses are absolute val. symbols) (5,2)

Q. When you’re trying to find the width do you disregard negative answers?
Ans: yes

Q. When the number under the radical is large should you trust that it’s simplified fully?
Ans: No (it’s probably not fam)

Q. What should you do any time you solve by squaring?
Ans: Plug the answers back in (to double-check)

Q. How do you find the domain of an equation with a radical?
Ans: Set the value under the radical equal to 0. IGNORE values outside the radical. (The domain ONLY includes values that make the numbers under the radical greater than or equal to 0)

Q. What helps when you’re trying to simplify a radical with a power higher than 2?
Ans: Writing out a table with perfect roots of the power

Q. What is the range of an inverse function?
Ans: The domain of the original function

Q. When finding the domain of composition functions should you find the domain of the simplified function or the unsimplified function?
Ans: The unsimplified function (ESPECIALLY if there are radicals)

Q. Should you leave negatives in the denominator?
Ans: no (multiply by everything negative 1)

Q. log_a c = b if
Ans: a^(b) = c

Q. log_6 36 = 2
Ans: 6^2 = 36

Q. log
Ans: log_10

Q. What do you do for exponential growth or decay problems?
Ans: initial x percent^time

Q. log_a (xy)
Ans: log_a x + log_a y

Q. log_a(x/y)
Ans: log_a x – log_a y

Q. log_a x^r
Ans: rlog_a x

Q. Should you round down when a problem asks for the smallest whole number?
Ans: No (round up to the next whole number regardless of the decimal size- it can be less than .5 and you’d still round up)

Q. If two logs with the same base are on different sides of an equation can you cancel the logs?
Ans: Yes (queen:3)

Q. Can you take the log of a negative number?
Ans: NO

Q. What do you do when solving complex log/exponential equations?
Ans: distribute and factor

Q. What is -log when graphing?
Ans: A reflection about the x-axis (multiply y coordinates by negative 1)

Q. What do you need to change when graphing log transformations that involve a change in x coordinate?
Ans: The domain and vertical asymptote

Q. What is the vertical asymptote of a normal log equation?
Ans: the y axis

Q. How do you complete the square?
Ans: Divide the middle coordinate by 2, square it (This is the third coefficient) , ADD THIS TO THE OTHER SIDE OF THE EQUATION AS WELL

Q. What is the equation of a circle
Ans: (x-h)^2 + (y-k)^2 = r^2

Q. What is the center of the circle (x-h)^2 + (y-k)^2 = r^2
Ans: (h,k) (h -> -h)

Q. What is the radius of the circle (x-h)^2 + (y-k)^2 = r^2
Ans: r

Q. You graph negative angles in standard form
Ans: clockwise

Q. You graph positive angles in standard form
Ans: counterclockwise

Q. How do you find the value of an angle?
Ans: Find its coordinate on the unit circle

Q. What are the Pythagorean identities?
Ans:
sin^2 + cos^2 = 1
tan^2 + 1 = sec^2
cot^2 + 1 = csc^2

Q. Arc length formula
Ans: s=r(theta)

Q. isosceles triangle
Ans: two congruent sides

Q. polygon
Ans: closed figure with no crossing / curved lines

Q. regular polygon
Ans: all sides and angles are equal

Q. pentagon
Ans: 5

Q. hexagon
Ans: 6

Q. heptagon
Ans: 7

Ans: four sided polygon

Q. trapezoid
Ans: one pair of parallel sides

Q. rhombus
Ans: A parallelogram with four congruent sides

Q. rectangle
Ans: A parallelogram with four right angles

Q. square
Ans: A parallelogram with four congruent sides and four right angles

Q. central angle
Ans: An angle whose vertex is the center of the circle

Q. chord
Ans: A segment whose endpoints lie on a circle

Q. (3-d figures) face
Ans: flat surface

Q. (3-d figures) edge
Ans: intersection of two faces

Q. (3-d figures) vertex
Ans: intersection of three edges

Q. prism
Ans: 3-d figure with parallel / congruent faces

Q. volume
Ans: The amount of space an object takes up

Q. volume equation
Ans: lwh

Q. (volume formula) cube
Ans: side^3

Q. (volume formula) triangle
Ans: (1/2bh)h

Q. (volume formula) pyramid
Ans: 1/3(h)(area of base)

Q. (volume formula) cylinder
Ans: (pi)r^2h

Q. (volume formula) cone
Ans: 1/3(pi)r^2h

Q. (volume formula) sphere
Ans: 4/3(pi)r^3

Q. slope formula
Ans: y2-y1/x2-x1

Q. no slope
Ans: zero on bottom

Q. zero slope
Ans: 0/6

Q. slope-intercept form
Ans: y=mx+b

Q. point-slope form
Ans: y-y1=m(x-x1)

Q. horizontal slope
Ans: m=0

Q. vertical slope
Ans: undefined

Q. direct variation formula
Ans: y=kx

Q. how to: direct variation
Ans: solve for k, plug k back into direct variation formula, plug in x / y values and solve

Q. vertex form
Ans: y-k=a(x-h)^2, vertex = (h,k)

Q. (vertex form) y=ax^2
Ans: vertex is (0,0)

Q. (vertex form) y=a(x-h)^2
Ans: vertex is (h,0)

Q. (vertex form) y-k=ax^2
Ans: vertex is (0,k)

Q. Axis of symmetry equation
Ans: x=h

Q. Axis of symmetry
Ans: The line that divides graph into two congruent reflected halves

Q. sum and product of roots
Ans: 0=x^2-(sum of roots) + (products of roots)

Q. minimum value
Ans: smallest y value

Ans: x = -b ± √(b² – 4ac)/2a

Q. 45-45-90 triangle equal sides
Ans: x

Q. 45-45-90 triangle hypotenuse
Ans: (√2)x

Q. 30-60-90 triangle short side
Ans: x

Q. 30-60-90 triangle-long side
Ans: x√(3)

Q. 30-60-90 triangle hypotenuse
Ans: 2x

Q. i^2
Ans: -1

Q. i^3
Ans: -i

Q. i^4
Ans: 1

Q. distance formula
Ans: d = √[( x₂ – x₁)² + (y₂ – y₁)²]

Q. midpoint formula
Ans: (x₁+x₂)/2, (y₁+y₂)/2

Q. equation of a circle
Ans: (x-h)²+(y-k)²=r²

Q. Permutation
Ans: of ways something can be ordered

Q. permutation equation
Ans: n!/(n-r)!

Q. combination
Ans: A grouping of items in which order does not matter

Q. combination equation
Ans: n!/r!(n-r)!

Q. change of base formula
Ans: logaX=logbX/logbA

Note: Math questions in ALEKS are prepared by mathematics professors whose areas of expertise give the program the ability to ask questions that are both challenging and beneficial to the learning process.

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