UC DAVIS ; MATH PLACEMENT WITH COMPLETE SOLUTIONS LATEST VERSION] UPDATED | 2024

UC DAVIS ; MATH PLACEMENT WITH COMPLETE SOLUTIONS LATEST VERSION] UPDATED | 2024 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