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