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
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
