Rules for Exponents

Many important rules must be applied when dealing with exponents.

• The most important rule is that when multiplying or dividing number with exponents, make sure they have the same base.                                                                                                           Ex: x^2 and x^6 can be multiplied, but not x^2 times z^4

Other important rules include:

1. When multiplying: add the exponents. Ex: x^2 times x^3= x^3+2 ,or x^5
2. When dividing: subtract the exponents. Ex: x^7/x^3= X^7-3, or x^4
3. When in parenthesis: multiply the exponents. Ex: (x^2)^3= x^2x3, or x^6
4. When exponent is negative: create a fraction with 1 as the numerator and the base to the positive version of the exponent to the denominator. Ex: x^-5 = 1/x^5
5. When the exponent equals 1: the final answer is the number of the base. Ex: 8^1=8
6. When the exponent equals zero: the answer is always one. Ex: 4^0=1

This picture clearly displays the rules I have explained.

Dimensions of a Matrix

The Dimension of a Matrix is the way of describing the different sizes of matrices. 


It is written as the number of rows by (x) the number of columns.


Rows are horizontal and Columns are vertical.

For example:  

As shown in the picture, there are 5  horizontal groups of numbers, and two vertical groups of numbers. This means that the dimension of this matrix is 5x2

Systems of Equations

There are 3 types of systems of equations:

1. Consistent Independent
2. Consistent Dependent
3. Inconsistent

• Consistent Independent

This system of equations is made up of two equations that intersect and have one solution. A system can automatically be determined consistent independent if the equations have different slopes and different y interecpts. Example: y = 2x+3

                                                      y = 4x+2         


In the picture, the solution is the point where the two lines cross. They seem to interesct at (1,2) which would be the one solution to this consistent independent equation.

• Consistent Dependent

 In this type of system, the solution is all real numbers. This is because the lines overlap, so they have all of the same solutions. They overlap because they have the same slope and the same y intercept. Because of this, it is very easy to identify equations that make up a consistent dependent system. Example: y = 2x+3
                   y = 2x+3

In this picture, the two colors show that there are two lines on top of each other, clearly indicating a consistent dependent system. In this example, the system is -3x +5

• Inconsistent

In this type of system, the equations have no solutuions. This is because they have the same slope and different y intercepts, making the lines parallel. Example: y = 6x+1
                                                                                               y = 6x+7

In this picture, the lines are inconsistent because of having the same slope and different y intercepts. As shown, the system has no solutions. The green line's equation is seems to be 2x+ 5 while the red line's equation is 2x-2


These are the 3 types of systems of equations! And here are examples of all 3 once more: