BASIC ALGEBRA REVIEW
Assume that letters such as a, b, x, y re present 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 nonrepeating 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 nonnegative). 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 viceversa; 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
; quotientsum/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 differenceof squares method
applies twice. 
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