UC DAVIS ; MATH PLACEMENT WITH COMPLETE SOLUTIONS 100% 2023/2024 latest update

UC DAVIS ; MATH PLACEMENT WITH COMPLETE SOLUTIONS 100% 2023/2024 latest update properties of exponents - whole number exponents: b^n = b • b • b... (n times) - zero exponent: b^0 = 1; b ≠ 0 - negative exponents: b^-n = 1/(b^n); b ≠ 0 - rational exponents (nth root): ^n√(b) = 1/(b^n); n ≠ 0, and if n is even, then b ≥ 0 - rational exponents: ^n√(b^m) = ^n√(b)^m = (b^(1/n))^m = b^(m/n); n ≠ 0, and if n is even, then b ≥ 0 operations with exponents - multiplying like bases: b^n • b^m = b^(n + m) (add exponents) - dividing like bases: (b^n)/(b^m) = n^(n-m) (subtract exponents) - exponent of exponent: (b^n)^m = b^(n • m) (multiply exponents) - removing parenthesis: > (ab)^n = a^n • b^n > (a/b)^n = (a^n)/(b^n) - special conventions: > -b^n = -(b^n); -b^n ≠ (-b)^n > kb^n = k(b^n); kb^n ≠ (kb)^n b^n^m = b^(n^m) ≠ ((b^n)^m) log basics - logb(1) = 0 - logb(b) = 1 inverse properties of logs - logb(b^x) = x - b^(logb (x)) = x laws of logarithms - logb(x) + logb(y) = logb ( x • y) - logb(x) - logb(y) = logb(x/y) - n • logb(x) = logb (x^n) distributive law ax + ay = a(x + y) simple trinomial x^2 + (a + b)x + (a • b) = (x + a)(a + b) difference of squares - x^2 - a^2 = (x - a)(x + a) - x^4 - a^4 = (x^2 - a^2)(x^2 + a^2) = (x - a)(x + a)(x^2 + a^2) sum or difference of cubes - x^3 + a^3 = (x + a)(x^2 - ax + a^2) - x^3 - a^3 = (x - a)(x^2 + ax + a^2) factoring by grouping acx^3 + adx^2 +bcx + bd = ax^2(cx + d) + b(cx + d) = (ax^2 + b)(cx + d) quadratic formula x = (-b ± √(b² - 4ac))/2a adding fractions find a common denominator ; a/b + c/d = a/b(d/d) + c/d(b/b) = (ad + bc)/bd subtracting fractions find a common denominator ; a/b - c/d = a/b(d/d) - c/d(b/b) = (ad - bc)/bd multiplying fractions (a/b)(c/d) = ac/bd dividing fractions - invert and multiply ; (a/b)/(c/d) = a/b • d/c = ad/bc canceling fractions - ab/ad = b/d - (ab + ac)ad = (a(b + c))/ad = (b + c)/d rationalizing fractions - if the numerator or denominator is √a , multiply by √a/√a - if the numerator or denominator is √a - √b, multiply by (√a + √b)/(√a + √b) - if the numerator or denominator is √a + √b, multiply by (√a - √b)/(√a - √b) first degree equations solved using addition, subtraction, multiplication, and division second degree equations solved by factoring or the quadratic formula absolute value equivalent to two equations without the absolute value sign > e.g. |x + 3| = 7 → +(x + 3) = 7 or -(x + 3) = 7 solving linear inequalities can be treated like a linear equation, however, when multiplying or dividing both sides of an inequality by negative numbers requires the inequality sign to be reversed solving absolute value inequalities - with absolute value inequalities, you will always have two problems to solve - solve everything outside the parenthesis before the inside after splitting the equation > e.g. 2(x - 5) > 10 → x - 5 > 5 → x > 10