GMAT Algebra
Calculate: (1/4)⁻⁴
(4/1)⁴ 4⁴ 256
Simplify: (3/4)⁻²
(4/3)² 4²/3² 16/9
Simplify: (5²⁵)^(5²⁵)
(5²⁵)^(5²⁵) (5)^(5²)^(5²⁵) (5)^(5²⁷)
Calculate: (4/9)⁻³/²
(9/4)³/² √(9/4)³ (√(9/4))³ (3/2)³ 3³/2³ 27/8
Formula: The sum of the numbers in any evenly spaced sequence
(First term + Last term)/2*Number of terms
Simplify: (a*b)/(c/d)
(a*b)*(d/c)
Simplify: a²-2ab+b²
(a-b)(a-b)=(a-b)²
Simplify: x²+2xy+y²
(x+y)(x+y)=(x+y)²
Simplify: x²-y²
(x+y)(x-y)
Simplify: (x²+x-12)/(x-2)=0
(x-3)(x+4)/(x-2)=0 x=3, x=-4 If either of the factors in the numerator is 0, then the entire expression equals 0.
Simplify: (x^y)*(z^y)
(xz)^y
Simplify: (y/z)⁻⁴
(z/y)⁴ OR z⁴/y⁴
What can you determine from the statement below? |x+b|<c
-b is the centerpoint of the inequality when charted on the number line
10³
1,000
How-to: Solve for an equation when the variable is in the exponent
1. Simplify both sides of the equation so the bases match 2. Drop the base 3. Solve for the variable
Memorize: .1²
1/100 OR .01
Memorize: .1³
1/1000 OR .001
³√1,000
10
4⁵
1024
√121
11
Simplify: 12!11!+11!10!
11!(12!+10!) 11!10!(12*11+1) 11!10!(132+1) 11!10!(133)
11²
121
5³
125
2⁷
128
√196
14
√225
15
2⁴
16
√256
16
13²
169
√289
17
Simplify: 17 ⁿ√289
17(289)^(1/n) 17(17²)^(1/n) 17*17^(2/n) 17^(2/n+1)
√324
18
√361
19
14²
196
1.4²
2
³√8
2
√400
20
6³
216
15²
225
3⁵
243
√625
25
16²
256
2⁸
256
4⁴
256
3³
27
17²
289
1.7²
3
³√27
3
√900
30
2⁵
32
18²
324
7³
343
19²
361
Simplify: 3⁸+3⁷-3⁶-3⁵
3⁵(3³+3²-3¹-1) 3⁵(27+9-3-1) 3⁵(32) 3⁵(2⁵) 6⁵
³√64
4
7²
49
Calculate: What is n? 4⁸+4⁸+4⁸+4⁸=4ⁿ
4⁸(1+1+1+1)=4ⁿ 4⁸(4)=4ⁿ 4⁹=4ⁿ n=9
2.25²
5
³√125
5
2⁹
512
8³
512
³√216
6
Simplify: 6/√6
6/√6 *√6/√6 (6√6)/6 1√6 √6
25²
625
5⁴
625
2⁶
64
4³
64
³√343
7
√49
7
9³
729
Simplify: 75^y27^(2y+1)=5⁴3ⁿ
75^y27^(2y+1)=5⁴3ⁿ (3*5²)^y * (3³)^(2y+1)=5⁴3ⁿ 3^y * 5^2y * 3^(6y+3)=5⁴3ⁿ 5^2y * 3^(7y+3)=5⁴3ⁿ
2³
8
³√512
8
3⁴
81
³√729
9
30²
900
Data Sufficiency: Is x<y? 1. x³<y³ 2. (x+y)(x-y) < 0
A
Define: Discriminant
A component of the quadratic formula that determines the number of possible solutions to a quadratic equation. A discriminant greater than 0 means there are 2 solutions A discriminant equal to 0 means there is one solution A discriminant less than 0 means there are no solutions
Data Sufficiency: Is x²y⁵z>0 1. (xz)/y > 0 2. y/z < 0
B For this statement to be true: 1. x cannot be 0 2. y and z need to be both positive or both negative Statement 2 tells us that y/z have opposite signs so x²y⁵z≤0
Calculate: Solve for y x-y>-5 x-2y<-7
Combination x-y>-5 * -1 -x+y<5 x-2y<-7 -y<-2 * -1 y>2
What can you determine from the statement below? xy<0
Either x is negative and y is positive OR x is positive and y is negative
Simplify: 4/(3-√2)
Multiply by 1, using its conjugate 4/(3-√2) * (3+√2)/(3+√2) (4(3+√2))/((3-√2)(3+√2)) (x+y)(x-y)=x²-y² (4(3+√2))/(3²-√2²) (4(3+√2))/(9-2) (12+4√2)/7
Is the statement sufficient to answer the question? Is zp negative pz⁴<0
No, because while this statement tells us that p is negative, it does not indicate whether or not z is negative or positive
Define: Perfect square quadratic equations
Quadratic equations with only one solution. (x+y)² (x-y)² e.g.: x²+8x+16=0 (x+4)(x+4)=0 (x+4)²=0 x=-4 x²-6x+9=0 (x-3)(x-3)=0 (x-3)²=0 x=3
Define: Range (Statistics)
The difference between the greatest possible output and the least possible output
Define: Domain
The possible inputs
Define: Range (of a function)
The possible outputs
Simplify: even√negative
Undefined
Data Sufficiency: If -1/3 ≤ x ≤ -1/5 AND -1/2 ≤ y ≤ -1/4 What is the least value of x²y?
What is the least value of x²y -1/18
Formula: Discriminant
b²-4ac If ax²+bx+c=0
Formula: The sum of the first n positive natural numbers
n=(n*(n+1))/2
What can you determine from the statement below? xy>0
x and y are either both positive or both negative AND x/y>0
Simplify: x³+2x²-3x=0
x(x²+2x-3)=0 x(x+3)(x-1)=0 x=0, x=-3, x=1
What can you determine from the statement below? (x+y)/(x-y)=1
x+y=x-y x+y-x=-y y+y=0 2y=0 y=0
What can you determine from the statement below? x²-y²>0
x+y>0 x-y>0
What can you determine from the statement below? (-x)⁹>0
x<0
What can you determine from the statement below? x²=x
x=1 OR x=0
Simplify: x^(a+b)/x^b
x^a
Simplify: (x^y)^z
x^yz
What can you determine from the statement below? x²-x<0
x²<x ∴ 0<x<1
Simplify: y = (4/m)/((1/m)+(2/x))
y = (4/m)/((1/m)+(2/x)) * (mx)/(mx) (4mx/m)/((1mx/m)+(2mx/x) 4x/(1x+2m)
Formula: Directly proportional equations
y/x=k x-input value y-output value k-proportionality
Simplify: x>r and y<s
y<s * -1 -y>-s +x>r x-y>r-s
What can you determine from the statement below? 0<x<1 and y=x⁷
y<x
What can you determine from the statement below? If 0<x<1 and y=x²
y<x
Formula: Inversely proportional equations
y=k/x x-input value y-output value k-proportionality
What can you determine from the statement below? -1<x<0 and y=x⁵
y>x
What can you determine from the statement below? -1<x<0 and y=x⁶
y>x
√2
~1.4
√3
~1.7
√5
~2.25
Simplify: √x²
√+x and √-x ±x
Simplify: √y³
√y²*√y y√y
