Download Algebraic Formula Cheat Sheet and more Cheat Sheet Algebra in PDF only on Docsity! Algebraic Formula Sheet Arithmetic Operations ac+ bc = c(a+ b) ( a b ) c = a bc a b + c d = ad+ bc bd a− b c− d = b− a d− c ab+ ac a = b+ c, a 6= 0 a ( b c ) = ab c a( b c ) = ac b a c − c d = ad− bc bd a+ b c = a c + b c( a b ) ( c d ) = ad bc Properties of Exponents xnxm = xn+m (xn)m = xnm (xy)n = xnyn x n m = ( x 1 m )n = ( xn ) 1 m ( x y )−n = ( y x )n = yn xn x0 = 1, x 6= 0 ( x y )n = xn yn 1 x−n = xn xn xm = xn−m x−n = 1 xn Properties of Radicals n √ x = x 1 n n √ xy = n √ x n √ y m √ n √ x = mn √ x n √ x y = n √ x n √ y n √ xn = x, if n is odd n √ xn = |x|, if n is even Properties of Inequalities If a < b then a+ c < b+ c and a− c < b− c If a < b and c > 0 then ac < bc and a c < b c If a < b and c < 0 then ac > bc and a c > b c Properties of Absolute Value |x| = { x if x ≥ 0 −x if x < 0 |x| ≥ 0 |xy| = |x||y| | − x| = |x| ∣∣∣x y ∣∣∣ = |x||y| |x+ y| ≤ |x|+ |y| Triangle Inequality |x− y| ≥ ∣∣∣|x| − |y|∣∣∣ Reverse Triangle Inequality Distance Formula Given two points, PA = (x1, y1) and PB = (x2, y2), the distance between the two can be found by: d(PA, PB) = √ (x2 − x1)2 + (y2 − y1)2 Number Classifications Natural Numbers : N={1, 2, 3, 4, 5, . . .} Whole Numbers : {0, 1, 2, 3, 4, 5, . . .} Integers : Z={... ,-3, -2, -1, 0, 1, 2, 3, .. .} Rationals : Q= { All numbers that can be writ- ten as a fraction with an integer numerator and a nonzero integer denominator, a b } Irrationals : {All numbers that cannot be ex- pressed as the ratio of two integers, for example√ 5, √ 27, and π} Real Numbers : R={All numbers that are either a rational or an irrational number} 1 Logarithms and Log Properties Definition y = logb x is equivalent to x = b y Example log2 16 = 4 because 2 4 = 16 Special Logarithms lnx = loge x natural log where e=2.718281828... log x = log10 x common log Logarithm Properties logb b = 1 logb b x = x ln ex = x logb 1 = 0 blogb x = x elnx = x logb (x k) = k logb x logb (xy) = logb x+ logb y logb ( x y ) = logb x− logb y Factoring xa+ xb = x(a+ b) x2 − y2 = (x+ y)(x− y) x2 + 2xy + y2 = (x+ y)2 x2 − 2xy + y2 = (x− y)2 x3 + 3x2y + 3xy2 + y3 = (x+ y)3 x3 − 3x2y + 3xy2 − y3 = (x− y)3 x3 + y3 = (x+ y) ( x2 − xy + y2 ) x3 − y3 = (x− y) ( x2 + xy + y2 ) x2n − y2n = (xn − yn) (xn + yn) If n is odd then, xn − yn = (x− y) ( xn−1 + xn−2y + ...+ yn−1 ) xn + yn = (x+ y) ( xn−1 − xn−2y + xn−3y2...− yn−1 ) Linear Functions and Formulas Examples of Linear Functions x y y = x linear function x y y = 1 constant function 2
