GRE Math equations

Fractions and Algebra — How do you add fractions with the same denominator?

a/b + c/b = (a+c)/b

Fractions and Algebra — How do you add fractions with different denominators?

a/b + c/d = (ad+bc)/bd

Fractions and Algebra — How do you subtract fractions with the same denominator?

a/b - c/b = (a-c)/b

Fractions and Algebra — How do you subtract fractions with different denominators?

a/b - c/d = (ad-bc)/bd

Fractions and Algebra — How do you multiply fractions?

a/b × c/d = ac/bd

Fractions and Algebra — How do you divide fractions?

a/b ÷ c/d = a/b × d/c = ad/bc

Fractions and Algebra — How do you find a reciprocal?

For a nonzero number x, the reciprocal is 1/x.

Fractions and Algebra — How do you compare positive fractions using cross-multiplication?

a/b > c/d when ad > bc (for positive denominators b and d).

Fractions and Algebra — How do you convert a fraction to a percent?

Divide numerator by denominator, multiply by 100, and add %.

Fractions and Algebra — How do you distribute division across a sum in the numerator?

(a+c)/b = a/b + c/b

Fractions and Algebra — How do you factor out a common factor?

ab + ac = a(b+c)

Fractions and Algebra — How do you square a fraction?

(a/b)^2 = a^2/b^2

Fractions and Algebra — How do you take the square root of a fraction?

√(x/y) = √x/√y for x ≥ 0 and y > 0.

Fractions and Algebra — What happens when 0 < x < 1 is squared?

x^2 < x < √x

Fractions and Algebra — What is the zero product property?

If ab = 0, then a = 0 or b = 0 (or both).

Fractions and Algebra — What is standard form of a quadratic equation?

ax^2 + bx + c = 0

Fractions and Algebra — How do you factor x^2 + bx + c?

Find p and q with p+q=b and pq=c; then (x+p)(x+q)=0.

Fractions and Algebra — How does x^2 - 3x - 28 factor?

(x-7)(x+4)

Fractions and Algebra — What does FOIL mean?

Multiply First, Outside, Inside, and Last terms of two binomials.

Fractions and Algebra — What is the square-of-a-sum identity?

(x+y)^2 = x^2 + 2xy + y^2

Fractions and Algebra — What is the square-of-a-difference identity?

(x-y)^2 = x^2 - 2xy + y^2

Fractions and Algebra — What is the difference-of-squares identity?

(x+y)(x-y) = x^2-y^2

Fractions and Algebra — What is the order of operations (PEMDAS)?

Parentheses; Exponents; Multiplication/Division left to right; Addition/Subtraction left to right.

Number Properties — What is odd + odd?

Even

Number Properties — What is even + even?

Even

Number Properties — What is even + odd?

Odd

Number Properties — What is odd × odd?

Odd

Number Properties — What is the product if at least one integer factor is even?

Even

Number Properties — What is the product or quotient of two numbers with the same sign?

Positive

Number Properties — What is the product or quotient of two numbers with different signs?

Negative

Number Properties — What are the primes less than 100?

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Number Properties — What is a factor?

An integer that divides another integer evenly.

Number Properties — What is a multiple?

A number times any integer.

Number Properties — What is the division algorithm?

Dividend = divisor × quotient + remainder; 0 ≤ remainder < divisor (positive divisor).

Number Properties — When is an integer divisible by 2?

When its last digit is 0, 2, 4, 6, or 8.

Number Properties — When is an integer divisible by 3?

When the sum of its digits is divisible by 3.

Number Properties — When is an integer divisible by 4?

When its last two digits form a number divisible by 4.

Number Properties — When is an integer divisible by 5?

When its last digit is 0 or 5.

Number Properties — When is an integer divisible by 6?

When divisible by both 2 and 3.

Number Properties — When is an integer divisible by 8?

When its last three digits form a number divisible by 8.

Number Properties — When is an integer divisible by 9?

When the sum of its digits is divisible by 9.

Number Properties — When is an integer divisible by 11?

When the alternating sum of its digits is divisible by 11.

Number Properties — How do you count positive factors from prime factorization?

If N = p^a × q^b × …, the number of positive factors is (a+1)(b+1)…

Number Properties — How do you find the least common multiple (LCM)?

Multiply all prime factors using the highest exponent appearing in any number.

Number Properties — How do you find the greatest common factor (GCF)?

Multiply shared prime factors using the lowest exponent appearing in each number.

Number Properties — How are LCM and GCF related for two positive integers?

LCM(a,b) × GCF(a,b) = ab

Number Properties — What is the units digit of n! for integer n ≥ 5?

0

Number Properties — How do you count trailing zeros in a product?

Count matched pairs of factors 2 and 5 in its prime factorization.

Number Properties — When does a reduced fraction have a terminating decimal?

When the denominator has no prime factors other than 2 and/or 5.

Number Properties — What is the units-digit cycle of powers of 2?

2, 4, 8, 6 (repeats every 4).

Number Properties — What is the units-digit cycle of powers of 3?

3, 9, 7, 1 (repeats every 4).

Number Properties — What is the units-digit cycle of powers of 4?

4, 6 (repeats every 2).

Number Properties — What is the units digit of a positive power of 5 or 6?

Powers of 5 end in 5; powers of 6 end in 6.

Number Properties — What is the units-digit cycle of powers of 7?

7, 9, 3, 1 (repeats every 4).

Number Properties — What is the units-digit cycle of powers of 8?

8, 4, 2, 6 (repeats every 4).

Number Properties — What is the units-digit cycle of powers of 9?

9, 1 (repeats every 2).

Number Properties — What characterizes a positive perfect square's prime factorization?

Every prime factor has an even exponent.

Number Properties — What characterizes a positive perfect cube's prime factorization?

Every prime factor has an exponent divisible by 3.

Number Properties — What is the GCF of consecutive integers?

1

Number Properties — What perfect squares from 0² through 15² should you memorize?

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Number Properties — What cubes from 0³ through 10³ should you memorize?

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Roots and Exponents — What are approximations for √2, √3, and √5?

√2 ≈ 1.4; √3 ≈ 1.7; √5 ≈ 2.2

Roots and Exponents — How do you multiply square roots?

√a × √b = √(ab) for nonnegative a and b.

Roots and Exponents — How do you divide square roots?

√a/√b = √(a/b) for a ≥ 0, b > 0.

Roots and Exponents — Can you distribute a square root across addition?

No. √(a+b) is generally not √a + √b.

Roots and Exponents — What is √(x²)?

|x|

Roots and Exponents — What is √((x+y)²)?

|x+y|

Roots and Exponents — What is the product rule for like-base exponents?

x^a × x^b = x^(a+b)

Roots and Exponents — What is the quotient rule for like-base exponents?

x^a/x^b = x^(a-b) for x ≠ 0.

Roots and Exponents — What is the power-of-a-power rule?

(x^a)^b = x^(ab) (where defined).

Roots and Exponents — What is the product rule for equal exponents?

x^a y^a = (xy)^a

Roots and Exponents — What is the quotient rule for equal exponents?

x^a/y^a = (x/y)^a for y ≠ 0.

Roots and Exponents — How do you express a radical as a fractional exponent?

The bth root of x^a equals x^(a/b), where defined.

Roots and Exponents — What is the value of x^0 for nonzero x?

1

Roots and Exponents — What is the value of x^1?

x

Roots and Exponents — What does a negative exponent mean?

x^(-n) = 1/x^n for x ≠ 0.

Roots and Exponents — What is the special repeated-addition exponent rule?

b copies of b^n add to b^(n+1); e.g., 2^n + 2^n = 2^(n+1).

Roots and Exponents — What happens when a positive number between 0 and 1 is raised to an integer power greater than 1?

The value gets smaller.

Roots and Exponents — What happens when a base greater than 1 is raised to a positive fractional exponent below 1?

The value is smaller than the original base but greater than 1.

Roots and Exponents — What are powers of 2 from 2^5 through 2^10?

32, 64, 128, 256, 512, 1024

Roots and Exponents — What are powers of 3 from 3^2 through 3^5?

9, 27, 81, 243

Roots and Exponents — What are powers of 4 from 4^2 through 4^4?

16, 64, 256

Roots and Exponents — What are powers of 5 from 5^2 through 5^4?

25, 125, 625

Roots and Exponents — How many final zeros appear in the square root of a perfect square ending in 2k zeros?

k zeros.

Roots and Exponents — How many decimal places appear in the square root of a perfect-square decimal with 2k decimal places?

k decimal places.

Roots and Exponents — How many final zeros appear in the cube root of a perfect cube ending in 3k zeros?

k zeros.

Absolute Value and Word Problems — What is the definition of absolute value?

If a ≥ 0, |a|=a; if a < 0, |a|=-a.

Absolute Value and Word Problems — How do you solve

expression| = k for k ≥ 0?|Set expression = k and expression = -k, then solve both.

Absolute Value and Word Problems — How do you solve (Broken A| = |B|?|Solve A=B or A=-B.)

A| = |B|?|Solve A=B or A=-B.

Absolute Value and Word Problems — What is the triangle inequality for absolute values?

|a+b| ≤ |a|+|b|

Absolute Value and Word Problems — What is the reverse triangle inequality?

|a-b| ≥ ||a|-|b||

Absolute Value and Word Problems — What is the total-cost-per-item formula?

Price per item = total cost / number of items.

Absolute Value and Word Problems — What is the profit formula?

Profit = revenue - total cost = revenue - (fixed costs + variable costs).

Absolute Value and Word Problems — What is the simple-interest formula?

Interest = principal × annual interest rate × time in years.

Absolute Value and Word Problems — What is the compound-interest formula?

A = P(1+r/n)^(nt); P principal, r annual rate, n compounding periods per year, t years.

Absolute Value and Word Problems — What is the linear-growth formula?

F_n = p + kn; p starting value, k increase per period, n number of periods.

Absolute Value and Word Problems — How can consecutive integers be represented?

x, x+1, x+2, …

Absolute Value and Word Problems — How can consecutive even integers or consecutive odd integers be represented?

x, x+2, x+4, … (choose x even or odd).

Absolute Value and Word Problems — How can consecutive multiples of 5 be represented?

5n, 5(n+1), 5(n+2), …

Absolute Value and Word Problems — What is the distance-rate-time equation?

Distance = rate × time.