# gmat algebra

 Simultaneous Equations: Solving by Substitution and Solving by Combination - subj. linear equation 1. line up the terms of the equations2. make one of the variables the same in both equations, then add or subtract3. add the equations to eliminate one of the variables4. solve the resulting equation for the unknown variable5. substitute into one of the original equations to solve for the second variable absolute value equations 1. isolate the expression within the absolute value brackets2. remove the absolute value brackets and solve the equation for 2 different casescase 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 01 raised to any power equals 1so if x = x2, 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 product103 = (2 x 5)3 = (2)3 x (5)3 = 8 x 125 = 1000(3x)4 = 34 x x4 = 81x4 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-24 is not equal to (-2)4-24 =-1 x 24 = -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 oneone exception is a base of 000 = 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(a5)4 = a5x4 = a20 fractional exponents the numerator tells you what power to raise the base to and the denominator tells you which root to take253/2=sqrt(253) = sqrt((52)3)) = 53 = 125 factoring out a common term if two terms with the same base are added or subtracted, you can factor out a common term113 + 114 = 113(1 + 11) = 113(12)ex. if x = 420 + 421 + 422, what is the largest prime factor of x?420 (1 + 41 + 42)420 (1 + 4 + 16)420 (21)420 (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, = lxlnot all equations with even exponents have 2 solutionsx2 = -9 --> no solutionx2 = 0 --> 0 odd exponents keep the sign of the base x3 = -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 equationex. solve the following equation for w: (4w)3 = 32w-1((22)w)3 = (25)w-126w = 25(w-1)eliminate the identical bases, rewrite the exponents as an equation, and solve6w = 5w - 5w = -5be 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 valueif 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 todenominator tells you which root to take simplifying roots you can only simplify roots by combining or separating them in multiplication and divisionyou cannot combine or separate roots in addition or subtractionyou can split up a larger product into its separate factors ( = X = 5X4 = 20similarly, you can also simplify two roots that are being multiplied together into a single root of the product imperfect squares the number 52 is an example of an imperfect square because its square root is not an integer disguised quadratics ex 3w2 = 6w3w2 - 6w = 0w(3w-6) = 0w = 0 or w = 2 perfect square in quadratic equation (z + 3)2 = 25sqrt both sidesabs (z + 3) = 5z + 3 = +/-5z = {2, -8} one-solution quadratics aka perfect square quadratics x2 + 8x + 16 = 0(x+4)(x+4)=0(x+4)2=0solution is -4 zero in the denominator: undefined math convention does not allow division by 0(x2+x-12)/(x-2)=0(x-3)(x+4)/x-2=0x cannot be equal to 2 the three special products x2 - y2 = (x + y)(x - y)x2 + 2xy + y2 = (x + y)(x + y) = (x + y)2x2 - 2xy + y2 = (x - y)(x - y)=(x - y)2common mistakes: (x + y)2 = x2 +y2, (x - y)2 = x2 - y2 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 termsthose 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 55-56 combining inequalities line up the variables, then combineit is not always possible to combine all the inequalitiesyou 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 barsabs(x+b)= to blank. if x is negative, x is <= to blankgraph it Authorjeffhn90 ID208164 Card Setgmat algebra Descriptionalgebra Updated2013-05-02T02:26:17Z Show Answers