Commutative Law For Addition: a + b = b + a ex. 5 + 7 = 7 + 5 =12 Commutative Law For Multiplication: a x b = b x a ex. 3 x 6 = 6 x 3 = 18 Associative Law For Addition: (a + b) + c = a + (b + c) ex. (4 + 6) + 5 = 4 + (6 + 5) = 15 Associative Law For Multiplication: (a x b) x c = a x (b x c) ex. (2 x 3) x 4 = 2 x (3 x 4) = 24 Distributive Law: a x (b +c) = (a x b) + (a x c) ex. 3 x (2 + 5) = (3 x 2) + (3 x 5)=21 

The additive inverse for a is (a) for any constant.or variable. a + (a) = 0 ex. 5 + (5)= 0 The mulitplicative inverse for a is 1/a for any constant or variable. a x 1/a = 1 ex, 5 x 1/5 = 1 The identity element for addition is 0 for any constant or variable. a + 0 = a ex, 5 + 0 = 5 The identity element for multiplication is 1 for any constant or variable. a x 1 = a ex, 5 x 1 = 5 

Distance Formula between two points on a cartesian graph Let the first point be (x_{1},y_{1}) and the second point be (x_{2},y_{2}). We use the Pythagorean thereom for a right triangle, using the length between the two x values as one side and the length between the y values as another side of a right triangle. The diagonal line, or shortest distance between the two points would be the hypotenuse of that right triangle. The distance, which we'll call z, between those two points would be: z^2 = (x_{2}  x_{1})^2 + (y_{2}  y_{1})^2 We solve for z by taking the square root of both sides. Sqrt is the same as square root here. We get z = sqrt[(x_{2}  x_{1})^2 + [(y_{2}  y_{1})^2 ] 
x^{m } times x^{n} = x ^{m + n} ex. 2^{3} x 2^{4} = 2^{3 + 4} = x^{7} x^{m}divided by x^{n} = x^{m  n} ex. 2^{7} / 2^{4} = 2^{7  4} = 2^{3} 
(x^{m})^{n} = x^{m^n} = x^{mn} ex. (x^{5})^{3} = x ^{5^3} = x^{15} 
(x^{m})^{1/n} = x ^{m/n} ex. (x^{15})^{1/3} = x^{5} 
ex. 4x  9 = 7 + 9x + 19 4x  9 = 26 + 9x Combine Like Terms. In the case, +9 = +9 add the 7 and the19. Add 9 to both sides to get rid of the 9 on the left hand side. 9 is the additive inverse of 9 and will cancel that term to zero. ^{} 4x + 0 = 35 + 9x 9x =  9x We need to get rid of the 9x on the right side now, so we'll subtract both sides by 9x. ^{} 5x = 35 (1)(5x) = (1)(20) Multiply both sides by 1 to get rid of the minus sign in front of the 5x. ^{} 5x = 35 (1/5)(5x) = (1/5)(35) Multiply both sides by 1/5, which is the same as dividing both sides by 5. We want to get rid of the 5 in front of the x to solve for x. x = 7 Our solution to the equation is x = 7, If you plug 7 into the original equation, the right side will equal the left side of the equation. 
x^{5}  2x^{4}  15x^{3} ^{} x^{6} + x^{5} + 12x^{4} (x^{3})(x^{2}  2x  15) Factor out the common factor ^{} of x^{3} from the numerator and (x^{4})(x x^{4}. 
polynomials in the numerator (x^{4})(x + 4)(x + 3) and the denominator. (x^{3})(x + 3)(x  5) Cancel out any common ^{} (x^{4})(x + 4)(x + 3) factors. x  5 We get the answer. ^{} x(x + 4) 