
Simultaneous Equations: Solving by Substitution and Solving by Combination  subj. linear equation
 1. line up the terms of the equations
 2. make one of the variables the same in both equations, then add or subtract
 3. add the equations to eliminate one of the variables
 4. solve the resulting equation for the unknown variable
 5. substitute into one of the original equations to solve for the second variable

absolute value equations
 1. isolate the expression within the absolute value brackets
 2. remove the absolute value brackets and solve the equation for 2 different cases
 case 1: x = z (x is positive)
 case 2: x = a (x is negative)
 3. check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides of the equation are in fact equal

dividing by 0 in a denominator by itself in the original equation  subj. linear equations
you are not allowed to divide by 0

base of 0 and 1  subj. exponents
 0 raised to any power equals 0
 1 raised to any power equals 1
 so if x = x^{2}, x must be either 0 or 1

a fractional base  subj. exponents
as the exponent increases, the value of the expression decreases

compound base  subj. exponents
 just as an exponent can be distributed to a fraction, it can also be distributed to a product
 10^{3} = (2 x 5)^{3} = (2)^{3} x (5)^{3} = 8 x 125 = 1000
 (3x)^{4} = 3^{4} x x^{4} = 81x^{4}

negative base  subj. exponents
 when dealing with negative bases, pay particular attention to PEMDAS.
 unless the negative sign is inside parentheses, the exponent does not distribute
 2^{4} is not equal to (2)^{4}
 2^{4 }=1 x 2^{4} = 16
 (2)^{4} = (1)^{4} x (2)^{4} = 1 x 16 = 16

multiplying exponents
when multiplying two exponential terms with the same base, add the exponents

dividing exponents
when dividing two exponential terms with the same base, subtract the exponents

exponents raised to the power of zero
 anything raised to the zero equals one
 one exception is a base of 0
 0^{0} = undefined

negative exponents
something with a negative exponent is just "one over" that same with a positive exponent

nested exponent: multiply exponents
 when you raise an exponential term to an exponent, multiply the exponents
 (a^{5})^{4} = a^{5x4 }= a^{20}

fractional exponents
 the numerator tells you what power to raise the base to and the denominator tells you which root to take
 25^{3/2}=sqrt(25^{3}) = sqrt((5^{2})^{3})) = 5^{3} = 125

factoring out a common term
 if two terms with the same base are added or subtracted, you can factor out a common term
 11^{3} + 11^{4} = 11^{3}(1 + 11) = 11^{3}(12)
 ex. if x = 4^{20} + 4^{21} + 4^{22}, what is the largest prime factor of x?
 4^{20} (1 + 4^{1} + 4^{2})
 4^{20} (1 + 4 + 16)
 4^{20} (21)
 4^{20} (3 x 7)
 now that you have expressed x as a product, you can see that 7 is the largest prime factor of x

even exponents hide the sign of the base
 for any x, = lxl
 not all equations with even exponents have 2 solutions
 x^{2} = 9 > no solution
 x^{2} = 0 > 0

odd exponents keep the sign of the base
x^{3} = 125 > 5

same base or same exponent
 if exponential expressions are on both sides of the equation, rewrite the bases so that either the same base or the same exponent appears on both sides of the exponential equation
 ex. solve the following equation for w: (4^{w})^{3} = 32^{w1}
 ((2^{2})^{w})^{3} = (2^{5})^{w1}
 2^{6w} = 2^{5(w1)}
 eliminate the identical bases, rewrite the exponents as an equation, and solve
 6w = 5w  5
 w = 5
 be careful if 0, 1, or 1 is the base (or could be the base), since the outcome of raising those bases to powers is not unique

Square root
 has one value
 if an equation contains a square root on the GMAT, only use the positive root (even root)
 odd roots  keep the sign of the base

roots and fractional exponents
 numerator tells you what power to raise the base to
 denominator tells you which root to take


imperfect squares
the number 52 is an example of an imperfect square because its square root is not an integer

disguised quadratics ex
 3w^{2} = 6w
 3w^{2}  6w = 0
 ^{}w(3w6) = 0
 w = 0 or w = 2

perfect square in quadratic equation
 (z + 3)^{2} = 25
 sqrt both sides
 abs (z + 3) = 5
 z + 3 = +/5
 z = {2, 8}

onesolution quadratics aka perfect square quadratics
 x^{2} + 8x + 16 = 0
 (x+4)(x+4)=0
 (x+4)^{2}=0
 solution is 4

zero in the denominator: undefined
 math convention does not allow division by 0
 (x2+x12)/(x2)=0
 (x3)(x+4)/x2=0
 x cannot be equal to 2

the three special products
 x^{2}  y^{2} = (x + y)(x  y)
 x^{2} + 2xy + y^{2} = (x + y)(x + y) = (x + y)^{2}
 x^{2}  2xy + y^{2} = (x  y)(x  y)=(x  y)^{2}
 common mistakes: (x + y)^{2} = x^{2} +y^{2}, (x  y)^{2} = x^{2}  y^{2}

formulas with unspecified amounts
apply the changes the question describes directly to the original expression

recursive sequences
 when a sequence is defined recursively, the question will have to give you the value of at least one of the terms
 those values can be used to find the value of the desired term

sequence problems/sequence and patterns
page 84

common squares and square roots + cubes and cube roots
page 5556

combining inequalities
 line up the variables, then combine
 it is not always possible to combine all the inequalities
 you can also add the inequalities together (you should never subtract or divide two inequalities; you can only multiply inequalities together under certain circumstances)
 only multiply inequalities together if both sides of both inequalities are positive

manipulating compound inequalities
must perform operations on every term in the inequality, not just the outside terms

inequalities and absolute value
 first, create a number line for the term inside the absolute value bars
 abs(x+b)<c
 the equation tells us that x must be exactly c units away from b

squarerooting inequalities
 recall that = abs(x)
 if x is positive, x is >= to blank.
 if x is negative, x is <= to blank
 graph it

