Math 10: Number Systems, Exponents, and Radicals Unit Final
integers (I)
-*Negative and positive whole* numbers -eg. I = ...-2, -1, 0, 1, 2...
Perfect square
-A number that's the product of an integer times itself -Eg. 5 ⋅ 5 = 5² = 25
Perfect cube
-A number that's the product of an integer times itself three times -Eg. 5 ⋅ 5 ⋅ 5 = 5³ = 125
Power
-A number written in exponential form -Consists of a base and an exponent -eg.
natural numbers (N)
-All *whole positive *counting numbers -The narrowest, most specific category -*Excludes 0* -eg. N = 1, 2, 3....
whole numbers (W)
-All *whole positive* counting numbers -*Includes 0* -eg. W = 0, 1, 2...
irrational numbers (Q￣)
-All non-terminating or repeating decimals -Eg. 𝝅, √5
Cube rooting
-Allows us to find the integer which is cubed to get a perfect cube -Used to determine if a number is a perfect cube
rational numbers (Q)
-Any number whose exact value is known -generally 4 types can classify as a ______ number
radical [expression]
-Any term that has a radical sign or a fractional exponent -has 2 types
Composite number
-Can be written as a* product of *prime numbers (*more than just 1 and itself*) -Is *not* a prime number
number system
-Gives us a way to organise and classify numbers based on their properties -Number system classification goes from most to least specific -Has 9 categories
prime numbers
-Numbers with only *1 and itself* as factors -1 is NOT a ____ ____
exponent
-Represents the repeated multiplication of a number by itself
complex/imaginary numbers (i)
-Square roots (or any roots) of negative numbers -Eg. -4 = 2i -When any number being rooted is a negative (like √-49), or is any [square] root that turns negative (like -√49)
Square rooting
-The "opposite" to squaring a number -Used to determine if a number is a perfect square
absolute value
-The absolute value of a real number, is the magnitude of the number (so you ignore the direction/sign) -Is a measure of a value's distance from zero -Is the square root of a squared number -eg.
dividing zero
-This leaves an unidentified set or the empty set of values -Eg. anything divided by zero
rationalizing decimals
-To rationalize a denominator means to make the bottom of the fraction (denominator) into a rational number (no radical). We will do this by multiplying the top and bottom of the expression by a certain radical -Case 1: monomial denominators (multiply the numerator and denominator by the denominator- remember: a radical multiplied itself will cancel out the radical sign) -eg. Case 2: Binomial denominators (you must multiply the numerator and denominator by the conjugate of the denominator- same binomial, opposite sign on second term) -eg.
mixed radical
-What the radical is called when the coefficient of the radical isn't equal to 1 -eg. FIXTHIS
rational exponenets
-When 25½ is multiplied to itself (which is squaring), the answer is 251. This means putting something to an exponent of ½, is the same as square rooting it. -eg. 25½ ⋅ 25½ = 251 is same as √25 ⋅ √25 = 25 -Exponents in the form of fractions represent radicals -The numerator is the exponent, and the denominator is what you're rooting the power by (aka a radical's index) -Eg. 25½ is same as √25 (1 is not written as an exponent- as usual- and neither is 2- if you're square rooting) FIXTHIS
conjugate pair
-When the binomials are multiplied together, the radicals in the first and last term are squared (so they cancel out each other's radical sign), and the middle terms are equal + opposite and cancel to zero. The result is a rational solution
entire radical
-When the coefficient of a radical is 1 (because the entire expression is under the radical sign) -eg. FIXTHIS
lowest common multiple (LCM)
DEFINITION -*is the smallest number that each number in a set can be multiplied by a whole number to achieve*AREYOUSUREBOUDIS -Must be greater than or equal to one/each number in the set (seperately, not added together) STEPS -Find the prime factors of each number -Any common factors are multiplied together (like when you're finding the GCF) -Value is then multiplied by all unused factors for all numbers in the set
real numbers (R)
DEFINITION -All natural, whole, rational and irrational numbers, and integers (*all types of numbers except for complex*) -all numbers are real unless they're the square root of a negative -eg. the number being rooted is a negative (like √-49), or is any square root that turns negative (like -√49) FORMULA -R = QUQ￣
greatest common factor (GCF)
DEFINITION -Is the *largest factor* of a set of whole numbers that *divides *into *each *number (in the set) *evenly.* -found by: *inspection or prime factorization* STEPS -Find the prime factors for each number in the set -Circle all common prime factors -multiply these together (from a single number in the set) = GCF
zero exponent law
EXPONENTS: -Any number with an exponent of 0 will equal to 1 -Eg. 5000 = 1 FIX THIS
power of a product law
EXPONENTS: -Eg. (axby)2 = ax2by2 FIXTHIS
power of a quotient law
EXPONENTS: -Eg. (axby)2 = ax2by2 FIXTHIS
negative exponent rule
EXPONENTS: -If the exponent is negative, changing the base to its reciprocal will make the exponent positive This method only changes the value of the exponent from positive to negative -so, all answers should be with positive exponents only -eg.
product law
EXPONENTS: -When a power is raised to an exponent, the 2 exponents are multiplied -Eg. (ax)y = ax⋅y
division law
EXPONENTS: -When powers with the same base are divided, the exponents are subtracted -eg.
multiplication law
EXPONENTS: -When powers with the same base are multiplied, exponents are added -Eg. ax ⋅ ay = ax+y FIX THIS
addition/subtraction law
EXPONENTS: -You can only add and subtract like terms (same base and exponent), but even then, only the coefficients are changed -eg. Eg. 2x3 + 3x3 = 5x3 FIXTHIS
-If a repeating decimal is a rational number, we should be able to convert from a repeating decimal to a fraction -Let "x" equal the repeating part of the decimal -Determine the length of period (repeating section) and multiply by: 10 if the period is 1 digit long/ 100 if the period is 2 digits long/ 1000 if the period is 3 digits long, and so on -Subtract the equations -Solve for x using reverse operations
List _ steps to convert repeating decimals into fractions.
-When you find the prime factors of a perfect square, each factor has a "pair" and taking 1 factor from each pair will give the square root. -When you find the prime factors of a perfect cube, each factor has a group of 3, and taking 1 factor from each group will give the cube root.
List _ steps to finding *perfect square roots and cubes* (using prime factorization.
STEPS: -Factor the largest perfect [square] possible out of the radicand Remove the perfect square (and square root it when you do so) Leave other factors under the radical sign -Find what perfect [square/cube/etc.] multiplied by a number equals the radicand The perfect [square] must be the largest possible [square/cube/etc.] root the perfect [square] and place it as a coefficient The number that can't be [square] rooted remains as the radicand If the radicand is an unknown variable (eg. x, y, etc.), then you find out how many groups of [2/3/etc.] are in the radicand. The number of groups becomes the exponent for the coefficient. -eg.
List _ steps to simplify radicals (aka: from any type of radical into the simplest mixed radical).
-Can be written as a fraction: m/n (where m and n are integers, and n is NOT 0) Fractions (F) F = {ab| a, bEI, b ≠ 0 } Terminating/repeating decimals Eg. Terminating: 0.4 Eg. Repeating: 0.33333.... Root of all numbers whose exact values are known Eg. 9, 25 All whole positive and negative numbers FIX THIS
List the 4 types that can classify as rational numbers.
-natural numbers (N) -whole numbers (W) -integers (I) -rational numbers (Q) -irrational numbers (Q￣) -real numbers (R) -complex/imaginary numbers (i)
List the 9 categories of the number system.
-entire -mixed
Name the 2 types of radicals.
8nb 8 = coefficient n = index (not included when square rooting by 2), what you're rooting something by b = radicand FIXTHIS
Name the _ parts of a radical.
converting from mixed into entire radicals
RADICALS: -Apply the index of the radical as an exponent of the coefficient and move this power inside the radical so that it is multiplied to the original radicand -Multiply the original radicand by the coefficient to the exponent of the index
multiply advanced radicals
RADICALS: -Case 1: monomials x binomials (apply the monomial to both binomials) -eg. -Case 2: binomial x binomial (using FOIL or adding like radicals then multiply) -eg.
converting from entire into mixed radicals
RADICALS: -Exactly the same as simplifying into mixed radicals
simplifying radicals with an index not equal to 2
RADICALS: -SAME AS SIMPLIFYING INTO MIXED RADICALS, or... PRIME FACTORIZATION: -List all the prime factors. The # of times 1 factor repeats, is the index. The factor that repeats is the coefficient If there are 2 repeating factors, you just multiply them to each other. The answer is the coefficient The remaining factor is the radicand -eg.
multiplying/dividing radicals
RADICALS: -We can multiply and divide radicals by applying each operation between coeffcients and between radicands (not between coefficients and radicands) -eg.
adding/subtracting radicals
RADICALS: -You can only add and subtract like radicals (make sure to simplify each radical completely before determining which radicals are "like" radicals) Only add and subtract the coefficients of the radicals for like terms. Do NOT change the radicand -eg.
