  # BASIC ALGEBRA REVIEW

Assume that letters such as a, b, x, y represent real numbers , unless otherwise specified.

 I. Subsets of the real numbers: Integers include whole numbers and their negatives. The rationals include all integers AND all fractions, and have repeating decimals . Irrationals (e.g., π , , etc.) include all other real numbers and have non-repeating decimals. A real number is either rational or irrational but not both. II. Order: a < b means that a is located to the left of b on the real number line ; a ≤ b means that either a is to the left of b, or a and b coincide. 0 is greater than all negative numbers, and less than all positive numbers . III. Absolute value: is the distance (on number line) from a to b, disregarding direction (so the result of an absolute value is always non-negative). If a > b , then ; else . If b = 0 , then is the distance from 0 to a. Watch negative signs closely; but . IV. Exponents: for a ≠ 0 . Change negative exponents to positive by exchanging the position of the associated factors from numerator to denominator or vice-versa; e.g., (note: do not exchange any factors that initially have positive exponents!). Other rules:  Fractional exponents are really radicals: V. Radicals (inverses of exponents): If then (read “the cube root of 216 equals 6”; note that the same three numbers 6, 3, 216 appear in both equations, but in a different order). The small 3 above the radical is the index ; the 216 inside the radical is the radicand. If the index is 2, it is usually not written; thus . Simplify radicals when possible by factoring; for a cube root, any group of 3 identical factors is moved outside the radical and compressed into one factor; for fourth roots, any group of 4 identical factors is moved and compressed, etc.; e.g., . VI. Operations involving radicals: Product ; quotient sum/difference (similar to combining like terms ); rationalizing denominators (one term) ; rationalizing denominators ( two terms , using conjugate of denominator) or ; if an expression combines different indices (e.g., a cube root times a fourth root), convert radicals to fractional exponents and use exponent rules to simplify (see IV above); result may be converted back to radicals afterwards. VII. Factoring polynomials: Simplify like terms (if any) and rewrite in descending-powers order. Then try methods in the order listed: (1) GCF ; (2) grouping (for four or more terms); (3) difference of squares, or sum/difference of cubes (for certain binomials); and (4) trinomial (“ac”) method. It may be necessary to apply multiple methods (or repeatedly apply a single method) within a single expression, e.g., ; the difference-of- squares method applies twice.
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