Reduction of a mixed number to an improper fraction. Reduction of an improper fraction to a mixed number. Addition and subtraction of fractions. Multiplication and division of fractions. Law of signs in division, changing signs of factors, etc. Complex fractions. ~Simultaneous Equations~ Solved by addition or subtraction. substitution. comparison. Graphical representation. ~Involution~ Law of signs. Binomial theorem laws. Expansion of monomials and fractions. binomials. trinomials. ~Evolution~ Law of signs. Evolution of monomials and fractions. Square root of algebraic expressions. Square root of arithmetical numbers. Optional Cube root of algebraic expressions. Cube root of arithmetical numbers. ~Theory of Exponents~ Proofs: a^m x a^n = a^(m + n); (a^m)/(a^n) = a^(m - n); (a^m)^n = a^(mn); [a^(mn)]^(1/n) = a^m; (a/b)^n = (a^n)/(b^n); (abc)^n = a^n b^n c^n. Meaning of fractional exponent. zero exponent. negative exponent. Four rules To multiply quantities having the same base, add exponents. To divide quantities having the same base, subtract exponents. To raise to a power, multiply exponents. To extract a root, divide the exponent of the power by the index of the root. ~Radicals~ Radical in its simplest form. Transformation of radicals Fraction under the radical sign. Reduction to an entire surd. Changing to surds of different order. Reduction to simplest form. Addition and subtraction of radicals. Multiplication and division of radicals a^(1/n) . b^(1/n) = [ab]^(1/n). ([ab]^(1/n))/(a^(1/n)) = b^(1/n). Rationalization Monomial denominator. Binomial denominator. Trinomial denominator. Square root of a binomial surd. Radical equations. _Always_ check results to avoid extraneous roots. ~Quadratic Equations~ Pure. x^2 = a. Affected. ax^2 + bx + c = 0. Methods of solving Completing the square. Formula. Developed from ax^2 + bx + c = 0. Factoring. Equations in the quadratic form. Properties of quadratics r_1 = -b/2a + ([b^2 - 4ac]^(1/2))/(2a). r_2 = -b/2a - ([b^2 - 4ac]^(1/2))/(2a). Then r_1 + r_2 = -b/a. r_1 . r_2 = c/a. Discriminant, b^2 - 4ac, and its discussion. Nature or character of the roots. ~Simultaneous Quadratics~ CASE I. One equation linear. The other quadratic. 3x - y = 12, x^2 - y^2 = 16. CASE II. Both equations homogeneous and of the second degree. x^2 - xy + y^2 = 21, y^2 - 2xy = -15. CASE III.