gmat algebra

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jeffhn90
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208164
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gmat algebra
Updated:
2013-05-01 22:26:17
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manhattan algebra
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algebra
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  1. 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
  2. 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
  3. dividing by 0 in a denominator by itself in the original equation - subj. linear equations
    you are not allowed to divide by 0
  4. base of 0 and 1 - subj. exponents
    • 0 raised to any power equals 0
    • 1 raised to any power equals 1
    • so if x = x2, x must be either 0 or 1
  5. a fractional base - subj. exponents
    as the exponent increases, the value of the expression decreases
  6. compound base - subj. exponents
    • just as an exponent can be distributed to a fraction, it can also be distributed to a product
    • 103 = (2 x 5)3 = (2)3 x (5)3 = 8 x 125 = 1000
    • (3x)4 = 34 x x4 = 81x4
  7. 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
  8. multiplying exponents
    when multiplying two exponential terms with the same base, add the exponents
  9. dividing exponents
    when dividing two exponential terms with the same base, subtract the exponents
  10. exponents raised to the power of zero
    • anything raised to the zero equals one
    • one exception is a base of 0
    • 00 = undefined
  11. negative exponents
    something with a negative exponent is just "one over" that  same with a positive exponent
  12. nested exponent: multiply exponents
    • when you raise an exponential term to an exponent, multiply the exponents
    • (a5)4 = a5x4 = a20
  13. fractional exponents
    • the numerator tells you what power to raise the base to and the denominator tells you which root to take
    • 253/2=sqrt(253) = sqrt((52)3)) = 53 = 125
  14. factoring out a common term
    • if two terms with the same base are added or subtracted, you can factor out a common term
    • 113 + 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
  15. even exponents hide the sign of the base
    • for any x,  = lxl
    • not all equations with even exponents have 2 solutions
    • x2 = -9 --> no solution
    • x2 = 0 --> 0
  16. odd exponents keep the sign of the base
    x3 = -125 --> -5
  17. 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: (4w)3 = 32w-1
    • ((22)w)3 = (25)w-1
    • 26w = 25(w-1)
    • 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
  18. 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
  19. roots and fractional exponents
    • numerator tells you what power to raise the base to
    • denominator tells you which root to take
  20. simplifying roots
    • you can only simplify roots by combining or separating them in multiplication and division
    • you cannot combine or separate roots in addition or subtraction
    • you can split up a larger product into its separate factors (=X = 5X4 = 20
    • similarly, you can also simplify two roots that are being multiplied together into a single root of the product
  21. imperfect squares
    the number 52 is an example of an imperfect square because its square root is not an integer
  22. disguised quadratics ex
    • 3w2 = 6w
    • 3w2 - 6w = 0
    • w(3w-6) = 0
    • w = 0 or w = 2
  23. perfect square in quadratic equation
    • (z + 3)2 = 25
    • sqrt both sides
    • abs (z + 3) = 5
    • z + 3 = +/-5
    • z = {2, -8}
  24. one-solution quadratics aka perfect square quadratics
    • x2 + 8x + 16 = 0
    • (x+4)(x+4)=0
    • (x+4)2=0
    • solution is -4
  25. 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=0
    • x cannot be equal to 2
  26. the three special products
    • x2 - y2 = (x + y)(x - y)
    • x2 + 2xy + y2 = (x + y)(x + y) = (x + y)2
    • x2 - 2xy + y2 = (x - y)(x - y)=(x - y)2
    • common mistakes: (x + y)2 = x2 +y2, (x - y)2 = x2 - y2
  27. formulas with unspecified amounts
    apply the changes the question describes directly to the original expression
  28. 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
  29. sequence problems/sequence and patterns
    page 84
  30. common squares and square roots + cubes and cube roots
    page 55-56
  31. 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
  32. manipulating compound inequalities
    must perform operations on every term in the inequality, not just the outside terms
  33. 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
  34. square-rooting inequalities
    • recall that  = abs(x)
    • if x is positive, x is >= to blank. 
    • if x is negative, x is <= to blank
    • graph it

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