# GMAT

 Simplify: XaXb Xa+b Simplify: AxBx (AB)x Simplify: Xa/Xb Xa-b Simplify: (A/B)x Ax/Bx (Ax)y Axy or (Ay)x X-a 1/(X)a Xa/b b√Xa Or b√(X)a A+xA+xAx 3Ax YOU CAN ONLY ADD OR SUBTRACT NUMBERS WITH EXPONENTS IF THE BASE AND EXPONENT ARE CONSTANT Ax+By CANNOT BE SIMPLIFIED ABx CANNOT BE SIMPLIFIED Simplify: (2+5)3 73 NOT 23+33 Simplify: (2*5)3 23*53 = 1000 X2-Y2 (X+Y)(X-Y) X2+2XY+Y2 (x+y)(x+y) OR (x+y)2 x2-2xy+y2 (x-y)(x-y) OR (x-y)2 X2-1 (X+1)(X-1) What are the Possible values for x ? (x-b)(x+a)/(x-m) = 0 +b,-a ZERO in the denominator means UNDEFINEDZERO in the numerator equals ZERO How do you calculate the SUM of a range of integers? ex. 20->100 Multiply the middle term(average) by the number of termsex. Average = (100+20)/2 = 60#of Terms = (100-20)+1 = 81Sum = 60*81 = 4860For any set where there is an ODD number of terms, the sum of all the integers is ALWAYS a multiple of the # of items (NOT TRUE if there is an Even number of terms) ex. 4860/81=60 What is the SUM of an EVENLY spaced Set? ex. 4,8,12,16,20 The sum of an evenly spaced set is the average times the number of termsex.Average = (20+4)/2 = 5#of terms = (20-4)/4+1 = 5 *after subtracting the highest and lowest digits, you need to divide by the increment b/w digits in the set before adding 1Sum = 12*5 = 60 How many integers are there in a consecutive set? ex. 14->765inclusive (Last-First)/increment + 1 = # of integers ex. (765-14)/1+1=752 How many integers are there in a consecutive set? ex. all even integers 12 to 24 inclusive ex. (24-12)/2+1=7 How many integers are there in a consecutive set? ex. multiples of 7 between 100 and 150inclusive ex. (147-105)/7+1=7Need to use the lowest and highest multiple of 7 within that range21*7 = 147 closest integer to 150 without going outside the range15*7 = 105 closest integer to 100 without going outside the range What is the Factor Foundation Rule? If A is a factor of B, and B is a factor of C, then A is also a factor of C. The product of any combination of factors of integer N is in an of itself a factor of integer N What is the forumal for DIRECT proportionality? ex. Height is the square of the velocity of an object, if an object is moving at 16ft/s reaches 4ft, what speed is required to reach 9ft? (Y1/X1) = (Y2/X2)ex. (4/162) = (9/v22) v22= 9(162/4) = 576√V = 24 What is the formula for INVERSE proportionality? ex. Current is inversley proportional to resistance, if a wire carries 4amps of current but the resistance decreases by 1/3 of its original value how many amps of current will flow? Y1X1 = Y2X2C1R1 = C2R24*3=C2*1 C2 = 2 Use patterns when finding terms in a sequence. ex. What is the units digit of S65 if Sn = 3n S1 = 31 = 3S2 = 32 = 9S3 = 33 = 27S4 = 34 = 81S5 = 35 = 243 65 is a multiple of 5, so it will share the same units digit of 3 3√1 1 3√8 2 3√27 3 3√64 4 3√125 5 13 1 23 8 33 27 43 64 53 125 √2 ~1.4 √3 ~1.7 √5 ~2.25 √121 11 √144 12 √169 13 √196 14 √225 15 √256 16 √625 25 n√xy (n√x)(n√y) n√(x+y) CANNOT BE SIMPLIFIED FURTHER n√(x/y) (n√x)/(n√y) Solve 4-3x<10 4-3x<10-3x<6 x>-2 REMEMBER when multiplying or dividing an inequality you need to FLIP the sign Solve for ab 1>1-ab>0 1>1-ab>00>1-ab>-10 90/6=15 What is the test to see if an integer is divisible by 8? If the integer is divisible by 2 THREE times or the last 3 digits are divisible by 8.ex118 128/2= 64 64/2 = 32 32/2=16 128/8=16 What is the test to see if an integer is divisible by 9? If the sum of the integers digits is divisible by 9ex 126 1+2+6=9 126/9=14 What is the lowest PRIME number? 2 What number is a MULTIPLE(more) of EVERY number? 0 What is the Greatest Common Factor (GCF)? The largest divisor of two or more integers. Calculated by multiplying ONLY the common factors a set of integers has.Ex. GCF of 30 and 24Factors of 30: 2 3 5Factors of 24: 2 2 2 3Common Factors: 2 3 = 2*3 = 6 What is the Least Common Multiple (LCM)? The smallest multiple of two or more integers. Calculated by multiplying all the primes of each integer but ONLY counting the common factors ONCE.Ex. LCM of 30 and 24Factors of 30: 2 3 5Factors of 24: 2 2 2 3Multiply Factors: 2*2*2*3*5 = 120 What is a remainder? The number LEFT OVER from long division.ex 3 5| 17 -15 2 is the remainder How can you pick a number with a desired remainder? If you need a desired remainder of N after division by X, pick a multiple of X and ADD Nex. Need a number with a remainder of 4 after division by 7, 14+4=18 2 7| 18 -14 4 is the remainder ODD+EVEN = ? ODD ODD-EVEN = ? ODD ODD+ODD = ? EVEN ODD-ODD = ? EVEN EVEN-EVEN = ? EVEN EVEN+EVEN = ? EVEN ODD*ODD = ? ODD ODD*EVEN = ? EVEN EVEN*EVEN = ? EVEN (also it will always be divisible by 4) If the sum of 2 primes is ODD, what must one of the primes be? 2 EVEN/EVEN = ? Can be EVEN, ODD, or NON-INTEGER EVEN/ODD = ? Can be EVEN, or NON-INTEGER not ODD ODD/EVEN Can ONLY be a NON-INTEGER, not ODD not EVEN ODD/ODD = ? Can ONLY be ODD or NON-Integer not EVEN How can you get the average of an evenly spaced set of integers? Take the average of the FIRST and LAST integer in the set How do you calculate the SUM of the terms in an evenly spaced set? Multiply the average by the #of terms. How can you find the #of terms between two integers? Subtract the LOWEST integer from the Highest, divide by the interval b/w each integer in the set and ADD 1.Ex.Set A: 6,7,8,9,10 : 10-6=4 +1 = 5integersSet B: 12,14,16,18,20,22,24 : 24-12=12 /2 = 6 +1 = 7integers If you have an ODD number of consecutive integers, will the average be an integer? YES - always The sum of an ODD number of consecutive integers will also ALWAYS be a multiple of the number of termsex. 4+5+6+7+8 = 30, a multiple of 5 If you have an EVEN number of consecutive integers, will the average be an integer? NO - never How do you calculate the sum of different parts of a set? Use N as the first integer and add the interval to each consecutive integer starting with N.ex. How much greater is the sum of the last four integers in a set of eight than the first 4 (n+4)+(n+5)+(n+6)+(n+7)-[ n +(n+1)+(n+2)+(n+3)] 4 + 4 + 4 + 4 = 16 , the last 4 intergers is 16 more than the first 4 integers 03 0zero to any power is zero 16 11 to any power is 1 170 1any number with 0 as the exponent is 1 1891 189Any number to the exponent of 1 is that number √(y3) √y√y√y = y√y REMEMBER variables with EVEN exponents always have TWO values, a negative and positive Variables with ODD exponents have ONE value, same sign as the base What is the MASTER RULE? If two equations are LINEAR (no squared terms) the equations WILL be sufficient to find the value of the variables unless the two equations are mathematically identical ex. x+y=12 2x+2y=24 if there are ANY NON-linear terms (X2,Y3,XY,X/Y) there are usually 2 or more values for each variable. Same base means same exponent ex. (4w)3=32dw-1((2)2)w)3 = (25)w-16W = 5(W-1)W = -5 What is the LINEAR FORMULA used for determining a value in a sequence? ex What is the rule for the sequence: 16,20,24,28 Sn = kn+xK = the constant interval between integers in a set Sn = 4n+xS1 = 4(1)+x16 = 4+xx = 12 What is the EXPONENTIAL FORMULA used for determining a value in a sequence?ex What is the rule for the sequence: 20,200,2000,20000 Sn = x(kn)K = is the constant that when multiplied by one number in the set yields the next Sn = x(10n)S1 = 2020 = x(101)x = 2 What is the formulat for LINEAR GROWTH y = mx+bm = constant rate of growth or decayb = y-intercept at time zerox = time How is Q related to each variable in the below formula: Q = 4W/4xz2 Q is DIRECTLY proportional to W, if one increases so does the otherQ is Inderectly proportional to x and z, if one increases the other decreases NEVER SUBTRACT OR DIVIDE INEQUALITIES, you can combine through multiplication or addition 8+LT2 LT10 or <10 8-LT2 GT6 or >6 remember toe FLIP the sign when subtracting extreme values 8*LT2 LT16 or <16 -7*LT2 GT(-14) remember to flip the sign when multiplying NEGATIVE extreme values 8/LT2 GT4 or >4 remember to FLIP the sign when DIVIDING extreme values LT2*LT8 LT16 or <16 What is the value(s) for X in |x-2|<5? |x-2|<5 absolute value formulas have 2 possible values, when the absolute value is negative or positivex-2<5 -(x-2)<5x<7 -x+2<5 x>-3-3