Math Concepts Flashcards
Prime Number
whole number whose only divisors are itself & 1
Composite number
one that has at least one other factor besides itself & 1
Proper fraction
a fraction whose numerator is less than its denominator
Improper fraction
fraction where the numerator is greater than or equal to the denominator
How to multiply two fractions?
multiply the two numerators and the two denominators
2/5 x 5/7 = 10/21 because 2x5=10 and 3x7=21
How to divide two fractions?
multiply the first fraction by the reciprocal of the second
27/4 ‘/’ 3/2 => 27/4 x 2/3 (reduces to) => 9/2 x 1/1 = 9/2 or 4 1/2
How to add fractions with same denominator?
to add fractions with the same denominator, add their numerators and preserve the denominator
3/4 + 1/4 = 4/4 = 1
How to add fractions with different denominator?
find a way to change them so that the denominators are the same
1/2 + 1/4 => (1/2 x 2/2 = 2/4) => 2/4 + 1/4 = 3/4
How to add fractions with very different denominators?
1/2 + 1/3 = (1/2 x 3/3) + (1/3 x 2/2) = 3/6 + 2/6 = 5/6
How to subtract fractions w/ same denominator?
keep the common denominator and subtract the second numerator from the first
3/4 - 1/4 = 2/4
How to subtract fractions with different denominators?
if different, change the fractions so that they have the same denominator
7/8 - 1/4 => 7/8 - (1/4 x 2/2 = 2/8) = 5/8
How to multiply decimals?
multiply them as if they were whole numbers, then add up their decimal places and insert a decimal point the same number of places from the right in the solution
How to divide decimals?
to divide by a whole number, all you have to do is keep the decimal point in the solution directly over the decimal point in the dividend (the number being divided)
How to convert fraction to decimal?
move the decimal two spaces to the right and add the percent sign
17% => 0.17
250% => 2.50
How to convert percent to decimal?
since percent means hundredths, you can convert percent to a decimal by dropping the percent sign and moving the decimal by dropping the percent sign and moving the decimal point two places to the left
- 5% => 0.055
- 3% => 0.003
How to convert fraction to a percent?
first change the fraction to a decimal then move the point two places to the right and add the percent sign
7/8 => .875 => 87.5%
What is the formula for simple interest?
i = p x r x t
i means interest or income
p means principal or money invested
r means rate of interest in percent
t means time (in years)
Absolute Value
the unsigned or undirected value of a number
|-3| = |3| = 3
What happens with the signs when you add?
even + even = even
odd + odd = even
even + odd = odd
odd + even = odd
What happens with the signs when you subtract?
even - even = even
odd - odd = even
even - odd = odd
odd - even = odd
What happens with the signs when you multiply?
pos x pos = pos
odd x odd = pos
pos x neg = neg
neg x pos = neg
Monomial
an algebraic expression with one and ONLY ONE term
2abfg | x^2y^4 | (2)(3)xyz | xyz/3
Binomial
an algebraic expression with two and ONLY TWO terms separated by either plus or minus signs
2x-3y | x^2-4y^2 | ab+4a^2 | abcd+5d
Trinomial
An algebraic expression with three and ONLY THREE terms separated by either plus or minus signs
x^2+2x-7 | 3x-4y+z
Polynomial
Any algebraic expression with MORE than ONE term.
2x+3y | abc-7d-8ce | 2x-3y-4z+4a-2b
Similar Terms
terms are similar or like when they have the SAME LITERAL FACTORS raised to the same powers
3x and 2x are like terms; 3x and 2x^2 are NOT
3x2 and -x^2 are like terms; 3x^2 and x are NOT
x^2z and -7x^2z are like terms; x^2z and -7xz are NOT
How to multiply algebraic expressions
multiply the numerical factors )also called coefficients) and then multiply the variables. When multiplying one power of x by another power of x, simply add the exponents.
(2x^2)(3x^5) => (2)(3) = 6 & (x^2)(x^5) = x^7 => 2+5 = 7
How to divide the algebraic expressions
divide the coefficient and then divide the variables. When dividing one power of x by another power of x simply subtract the exponents.
x^5/x^3 => 5-3 = 2 => x^2
8x^3/4x => (8/4 = 2)/(4/4 = 1) * (x^3 - x^1 = x^2)/(x/x = 1) = 2 * x^2 = 2x^2
How to factor?
factoring should be ONLY prime factors.
50 = (1)(50) is incorrect because neither 1 nor 50 is prime 50 = (2)(25) is incorrect because 25 is not prime 50 = (5)(10) is incorrect because 10 is not prime 50 = (2)(5)(5) is the CORRECT factorization because 2 and 5 are indeed prime numbers
How to find Greatest Common Factor
GCF –
75x^2 + 30x
1) find common factors of the numbers
75x^2 = 355xx
30x = 235*x
2) see what is in all of the numbers
3, 5, x
3) multiply these terms
35x = 15x
4) finish factoring
=15x(5x + 2)
Factoring the difference of two squares
the special product of the sum of two numbers and the difference of the same two numbers equals the difference of the squares of these numbers
(a + b)(a - b) = a^2 - b^2
Law of Exponents
x^0 = 1
x^-1 = 1/x x^-2 = 1/x^2 x^-3 = 1/x^3
x^a * x^b = x^a + b
x^a / x^b = x^(a - b) = 1 / x^b - a
x^a * y^a = (xy)^a
(x/y)^a = x^a / y^a
(x^a)^b - x^ab
Laws of Square Roots
√a^2=a
√a√b=√ab
√a / √b = √a / b
How to Foil
First Outer Inner Last - FOIL
positive L, positive O + I
x^2 + 4x + 3 = (x + )(x + )
positive L, negative O + I
x^2 - 4x + 3 = (x - )(x - )
negative L, negative O + I
x^2 - 4x - 3 = (x + )(x - )
negative L, positive O + I
x^2 + 4x - 3 = (x + )(x - )
Sum of angles in a polygon
The sum of ANY polygon can be found using: 180(n - 2)
n = number of sides
Triangle: 180(3 - 2) = 180 Quadrilateral: 180(4 -2) = 360 Pentagon: 180(5 - 2) = 540 Hexagon: 180(6 - 2) = 720 [...]
Special Right Triangles
Most common triangles:
(leg-leg-hypotenuse)
3-4-5 ;; 1-1-√2 ;; √3-1-2
30-60-90 triangles
The short side, opposite the 30° angle, is always half the hypotenuse. The longer leg is always the square root of 3 times longer than the shorter leg.
45-45-90
The two legs are always equal because this is an isosceles triangle, and the hypotenuse is always the square-root of two times any leg.
Rule of Exponents
Rule 1: x^-a = 1 / x^a
==> 4^-3 = 1 / 4^-3 = 1 / 64
==> 1 / 2^-a = 2^a
Rule 2: (x^a)(x^b) = x^a+b
==>(3^2)(3^4) = 3^(2+4) = 3^6 = 729
==> (y^3)(y^-1) = y^2
Rule 3: x^a / x^b = x^(a-b) = 1 / x^(b-a)
==> 5^7 / 5^4 = 5^(7-4) = 5^3 = 125
==> t^3 / t^8 = t^-5 = 1 / t^5
Rule 4: x^0 = 1
==>7^0 = 1
==> -3^0 = 1
NOTE: 0^0 is undefined
Rule 5: (x^a)(y^a) = (xy)^a
==> 2^3 * 3^3 = 6^3 = 216
==> (10z)^3 = 10^3z^3 = 1000z^3
NOTE: uses common exponents
Rule 6: (x / y)^a = x^a / y^a
==> (3 / 4)^2 = 3^2 / 4^2
==> (r / 4t)^3 = r^3 / 64t^3
Rule 7: (x^a)^b = x^ab
==> (2^5)^2 = 2^10 = 1024
==> (3y^6)^2 = (3^2)(y^6) = 9y^12
