uc davis ; math placement Questions &
Answers (GRADED A+)
properties of exponents - ANSWER- 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 - ANSWER- 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 - ANSWER- logb(1) = 0
- logb(b) = 1
inverse properties of logs - ANSWER- logb(b^x) = x
- b^(logb (x)) = x
laws of logarithms - ANSWER- logb(x) + logb(y) = logb ( x • y)
, - logb(x) - logb(y) = logb(x/y)
- n • logb(x) = logb (x^n)
distributive law - ANSWERax + ay = a(x + y)
simple trinomial - ANSWERx^2 + (a + b)x + (a • b) = (x + a)(a + b)
difference of squares - ANSWER- 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 - ANSWER- 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 - ANSWERacx^3 + adx^2 +bcx + bd = ax^2(cx + d) + b(cx + d) = (ax^2 + b)(cx + d)
quadratic formula - ANSWERx = (-b ± √(b² - 4ac))/2a
multiplying fractions - ANSWER(a/b)(c/d) = ac/bd
dividing fractions - ANSWER- invert and multiply ; (a/b)/(c/d) = a/b • d/c = ad/bc
canceling fractions - ANSWER- ab/ad = b/d
- (ab + ac)ad = (a(b + c))/ad = (b + c)/d
rationalizing fractions - ANSWER- 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)
adding fractions - ANSWERfind a common denominator ; a/b + c/d = a/b(d/d) + c/d(b/b) = (ad +
bc)/bd
Answers (GRADED A+)
properties of exponents - ANSWER- 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 - ANSWER- 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 - ANSWER- logb(1) = 0
- logb(b) = 1
inverse properties of logs - ANSWER- logb(b^x) = x
- b^(logb (x)) = x
laws of logarithms - ANSWER- logb(x) + logb(y) = logb ( x • y)
, - logb(x) - logb(y) = logb(x/y)
- n • logb(x) = logb (x^n)
distributive law - ANSWERax + ay = a(x + y)
simple trinomial - ANSWERx^2 + (a + b)x + (a • b) = (x + a)(a + b)
difference of squares - ANSWER- 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 - ANSWER- 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 - ANSWERacx^3 + adx^2 +bcx + bd = ax^2(cx + d) + b(cx + d) = (ax^2 + b)(cx + d)
quadratic formula - ANSWERx = (-b ± √(b² - 4ac))/2a
multiplying fractions - ANSWER(a/b)(c/d) = ac/bd
dividing fractions - ANSWER- invert and multiply ; (a/b)/(c/d) = a/b • d/c = ad/bc
canceling fractions - ANSWER- ab/ad = b/d
- (ab + ac)ad = (a(b + c))/ad = (b + c)/d
rationalizing fractions - ANSWER- 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)
adding fractions - ANSWERfind a common denominator ; a/b + c/d = a/b(d/d) + c/d(b/b) = (ad +
bc)/bd
