Arithmetic Integers: Any positive or negative whole number including zero Rules of integer calculations: Adding Same signs add and keep sign Different signs subtract absolute values and keep the sign of the number furthest from zero Subtracting: Apply the add opp property then follow the rules for addition. 8 + 12 = 20 same signs add and keep sign 12 + (12 ) 7 different signs subtract and keep the sign of the number furthest from zero 1 9 1 + ( 9) 24 apply add op property same signs for addition 8 ( ) 8 + (+) (8 ) apply add op property different signs for addition Multiplying and dividing integers: Rules are the same Same sign positive result; different signs negative result. In a series of integers, even number of same sign positive result and odd number of same sign, negative result
Absolute value a numbers distance from zero 4 = 4 This is a true statement because both numbers are four units from zero. When an absolute value expression has an arithmetic operation, perform that first before finding the absolute value and applying order of operations for the rest of the expression. + 4 + 6 8 4 + 6 8 4 + 6 4 + 6 10 original problem calculate inside absolute value find absolute value apply order of operations for the rest Factors and multiples: Prime factorization: Factoring a numbers until the only factors are prime. Greatest common factor (GCF): The largest number two or more numbers are divisible by. Using prime factorization is one way to find the GCF Least Common Multiple (LCM): smallest number that is divisible by two or more numbers. GCF using Prime Factorization 40 4 10 2 2 2 LCM Using Prime Factorization 70 2 2 7 180 9 20 2 2 20 4 2 2 GCF: 2 2 =20 20 is the largest factor of both 40, 180 LCM: 2 2 7 = 140 140 is the smallest multiple of both 20 and 70
Order of Operations: (PEMDAS) 1. Parentheses symbols of inclusion []; {}; division bar 2. Exponents. Multiplication and division work left to right as the operations occurs 4. Addition and subtraction work left to right as the operation occurs ( + ) 2 8 + 4 2 8 4 7 ( 8) 2 8 + 4 2 8 4 7 64 8 + 4 2 8 4 4 16 8 + 4 4 4 16 8 + 4 16 8 + 4 16 12 16 4 Original problem Symbols of inclusion: parentheses; above below bar Do mult/division L to R Since division occurs first do it before multiplication Add/sub from L to R Since subtraction occurs first begin there. Rules of divisibility
Fractions and Decimals Simplifying a fraction: Method A: Find the GCF of the numerator and denominator of the fractions. Divide both the numerator and denominator by the GCF Method B: Method A Use prime factorization for simplification by canceling common prime. Method B 6 60 6 12 60 12 6 60 2 2 2 2 Converting a decimal to an equivalent decimal: Multiply the numerator and the denominator with the same value Example: Convert to a fraction with a denominator of 4 6 6 9 9 = 4 4 Adding and subtracting fractions: Adding and subtracting fractions requires the same denominator. Find the LCM of the denominator then convert the fractions. Once converted, calculate the numerators, keep the denominator, and simplify the fractions if possible. 8 + 7 12 8 + 7 12 2 2 1 24 + 14 24 29 24 1 24
Multiplying fractions: Multiplying fractions you have two options: 1. Multiply numerator, denominators and then simplify 2. Short cut by canceling common factors from the numerator/denominator Direct multiplication 11 1 10 110 9 110 49 2 9 Short cut 11 1 10 11 1 10 2 9 Dividing fractions: Multiply the dividend (first fraction) by the reciprocal of the divisor (second fraction. Reciprocal is the multiplicative inverse of a number. The product of a number and its reciprocal is one. has a reciprocal of. 2 1 8 4 2 1 4 8 4 Comparing fractions: Convert the fractions by using the LCD (Least Common Denominator) or finding their cross products. Multiply the first numerator by the second fraction s denominator, then multiply the denominator of the first fraction by the numerator of the second fraction. 7 8 and 6 7 6 = 42 and 8 = 40. Since 42 > 40, the first fraction is larger
Converting a fraction to a decimal: Divide the denominator into the numerator by placing a decimal point behind the numerator s value and adding as many zero s as necessary. 8 8). 00 0.7 8).000 Converting a decimal to a fraction. Read the decimal using place value. The place value is the denominator and the number is the numerator. Simplify 0.4 is read 4 hundredths. Write as a fraction, then simplify. 4 100 4 100 9 20 Common percent equivalencies:
Exponents and roots: An exponent tells you how many times a numbers is multiplied by itself. A root of a number is the factor that is multiplied to get a specific quantity. Number Exponential form roots 27 81 4 or 9 2 or 9 2 2 A square root of a non negative number is a number that when multiplied by itself gives a specific quantity. The radical sign is a symbol used to indicate you are looking for the root of a number. The radicand is the number you are looking for the root of. The index tells you which root you are looking for. 4 169 16 27 The radicand is 169; the index is 2 never shown; You are looking for the square root of 169 which is 1. The radicand is 16; the index is 4. You are looking for the 4 th root of 16 which is 2. The radicand is 27; the index is. You are looking for the cube root of 27 which is. Adding and subtracting radicals When adding and subtracting radicals, you need to make sure that the radicands are the same then add or subtract the coefficient outside the radical sign. 8 + 6 ( 8 + 6) 14 Apply the distributive property, add the coefficients and keep the radical 2 + 2 6 2 + 4 (+ 6) 2 + 4 2 2 + 4 Apply the distributive property and calculate the coefficients of 2. Since is not the same it can t be simplified.
Simplifying radicals When simplifying radicals (square roots) factor the number into the largest perfect square factor and a non perfect square factor. Simplify by placing the root of the perfect square in front of the radical and the non perfect square remains inside the radical sign. 12 2 108 6 6 12 is not a perfect square but it is the product of a perfect square and another number. 108 is not a perfect square but it is the product of 6 and. Find the root of 6 and place outside the radical and leave the inside. Multiplying or dividing radicals: Basic rule: multiply (or divide) radicand by radicand and coefficient by coefficient, then simplify if possible. a b = ab a b = a b 2 6 ( ) 2 6 1 12 1 4 1 2 0 Group coefficients together and radicands together, then multiply. See if the radical can be simplified. In the case 12 is the product of a perfect square and another number. Extract the root of the perfect square and multiply it by the coefficient leaving the remaining factor in the radical sign. 6 1 2 Divide the coefficients then divide the radicals. Check for simplification. In this case you can t simplify