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|a+bi|

√[(a^2)+(b^2)]

Number of digits in x^y

(Ex. 7^43)

(Ex. 7^43)

y(logx)+1

43(log7)+1

36.339...+1

37.339...

37 (Eliminate the decimal part)

43(log7)+1

36.339...+1

37.339...

37 (Eliminate the decimal part)

"k"th Triangular Number

T(k)=[k(k+1)]/2

"n"th Pentagonal Number

P(n)=[n(3n-1)]/2

Sum of First "n" Perfect Squares

[n(n+1)(2n+1)]/6

Sum of First "n" Perfect Cubes

[(n/2)(n+1)]^2

What is the determinant of a singular matrix?

0

That is why singular matrices have no inverses.

That is why singular matrices have no inverses.

For any square matrix A,

A • A^(-1)

A • A^(-1)

The Identity Matrix

| 1 0 |

| 0 1 | For a 2x2 matrix,

| 1 0 0 |

| 0 1 0 |

| 0 0 1 | For a 3x3 matrix, etc.

| 1 0 |

| 0 1 | For a 2x2 matrix,

| 1 0 0 |

| 0 1 0 |

| 0 0 1 | For a 3x3 matrix, etc.

For any square matrix A,

|A| - |A^T|

(A^T is the transverse of the matrix A)

|A| - |A^T|

(A^T is the transverse of the matrix A)

0

The determinant of any square matrix is equal to the determinant of its transverse matrix.

The determinant of any square matrix is equal to the determinant of its transverse matrix.

Area of a Triangle Using Matrices

1 | x(1) y(1) 1 |

±— | x(2) y(2) 1 |

2 | x(3) y(3) 1|

1/2 is indicated as ± so that you can multiply whichever number would result in a positive value, depending on the situation, because area must be positive.

x(n) or y(n) means x "sub-n" or y "sub-n"

±— | x(2) y(2) 1 |

2 | x(3) y(3) 1|

1/2 is indicated as ± so that you can multiply whichever number would result in a positive value, depending on the situation, because area must be positive.

x(n) or y(n) means x "sub-n" or y "sub-n"

Arithmetic Mean of a and b

(a+b)/2

For a, b, and c, the arithmetic mean is

(a+b+c)/3 etc.

For a, b, and c, the arithmetic mean is

(a+b+c)/3 etc.

Geometric Mean of a and b

√(ab)

For a, b, and c, the geometric mean is

∛(abc) etc.

For a, b, and c, the geometric mean is

∛(abc) etc.

Harmonic Mean of a and b

(2ab)/(a+b)

For a, b, and c, the harmonic mean is

(3abc)/(a+b+c)

For a, b, and c, the harmonic mean is

(3abc)/(a+b+c)

Base Changing

Sorry, I don't know how to explain this one on here you just have to know it.

Number of Positive Integral Factors of a Number (Ex. 3240)

Find the prime factorization of the number and multiply the exponents of the prime factors after adding one to each of them.

3240 = 2^3 • 3^4 • 5

(3+1)(4+1)(1+1)

4 • 5 • 2

40

3240 = 2^3 • 3^4 • 5

(3+1)(4+1)(1+1)

4 • 5 • 2

40

Binomial Theorem

Sorry, again, I don't know how to explain this one on here you just have to know it.

Circular Permutations (keyring)

(n-1)!/2

Circular Permutations (non-keyring)

(n-1)!

√n+√(n+√(n+...=x

Square both sides

Set the radical junk equal to x

Solve the quadratic equation

The positive solution is the solution to the entire equation

Set the radical junk equal to x

Solve the quadratic equation

The positive solution is the solution to the entire equation

Find A+B / Find AB

√(8-2√15)=√A+√B

√(8-2√15)=√A+√B

Square both sides

8-2√15=A+B+2√AB

A+B=8

AB=15

8-2√15=A+B+2√AB

A+B=8

AB=15

Sum of the Roots of a Quadratic Equation

ax^2 + bx + c = 0

-b/a

-b/a

Product of the Roots of a Quadratic Equation

ax^2 + bx + c = 0

c/a

c/a

Sum of the Reciprocals of the Roots of a Quadratic Equation

ax^2 + bx + c = 0

-b/c

Sum of Roots/Product of Roots

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

-b/c

-b/c

Sum of Roots/Product of Roots

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

-b/c

Sum of the Squares of the Roots of a Quadratic Equation

ax^2 + bx + c = 0

(b^2 - 2ac)/a^2

(b^2 - 2ac)/a^2

Sum of the Squares of the Reciprocals (or the Reciprocals of the Squares) of the Roots of a Quadratic Equation

ax^2 + bx + c = 0

(b^2 - 2ac)/c^2

(b^2 - 2ac)/c^2

Sum of the Roots of Other Polynomial Equations

Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0

-B/A

-B/A

Product of the Roots of Other Polynomial Equations

Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0

If n is even, K/A

If n is odd, -K/A

If n is even, K/A

If n is odd, -K/A

Sum of the Reciprocals of the Roots of Other Polynomial Equations

Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0

-J/K

-J/K

Sum of the Squares of the Roots of Other Polynomial Equations

Ax^n + Bx^(n-1) + Cx^(n-2) +...+ Jx + K=0

(B^2 - 2AC)/A^2

(B^2 - 2AC)/A^2

Number of Zeroes at the End of a Specific Number Factorial

Ex. 1000

Ex. 1000

Divide the number by the largest integral power of 5 less than the original number

Continue dividing by decreasing integral powers of 5 until you reach 5

Add the quotients

Ex.

1000÷625=1

1000÷125=8

1000÷25=40

1000÷5=200

1+8+40+200=249

249 zeroes

Continue dividing by decreasing integral powers of 5 until you reach 5

Add the quotients

Ex.

1000÷625=1

1000÷125=8

1000÷25=40

1000÷5=200

1+8+40+200=249

249 zeroes

"p"th Even Perfect Number

(2^p - 1)•(2^(p-1))

Where p is a prime number

Where p is a prime number

In a rectangle with the dimensions x by y, the number of distinct paths from one corner to the other

(x+y)!/x!y!