Rules for Exponents

Whenever you multiply two terms with the same base, you can add the exponents.

When given a problem such as this: Simplify (x³)(x⁴) you must add the exponents, the powers given to the base numbers, giving you a final answer of x⁷ Adding the exponents when multiplying numbers with the same base is a rule.  

This is represented as: ( x ^m ) ( x ⁿ ) = x^{( m + n )}              

Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power

When given a problem such as: Simplify (x²)⁴   

You must multiply the exponents together, resulting in the final answer: x⁸

This is represented as:  ( x^m ) ⁿ = x ^{m n}

If you have a product inside parentheses, and a power on the parentheses, then the power goes on each element inside. For instance, (xy²)³ = (xy²)(xy²)(xy²) = (xxx)(y²y²y²) = (xxx)(yyyyyy) = x³y⁶ = (x)³(y²)³. Another example would be:

 

This rule does NOT work if you have a sum or difference within the parentheses. Exponents, unlike multiplication, do NOT "distribute" over addition.

Anything to the power zero is just "1"

Law	Example
x1 = x	61 = 6
x0 = 1	70 = 1
x-1 = 1/x	4-1 = 1/4
xmxn = xm+n	x2x3 = x2+3 = x5
xm/xn = xm-n	x6/x2 = x6-2 = x4
(xm)n = xmn	(x2)3 = x2×3 = x6
(xy)n = xnyn	(xy)3 = x3y3
(x/y)n = xn/yn	(x/y)2 = x2 / y2
x-n = 1/xn	x-3 = 1/x3

Dimensions of Matrices

The dimensions of a matrix can be determined by it's rows and columns.

The rows of a matrix are the numbers listed horizontally.

The columns of a matrix are the numbers listed vertically.

 

When you look at a matrix, the way you determine it's                                                                               dimensions is by first looking at the rows. Then you look

at the columns. In the example shown, there are 2 rows 

and 2 columns, making this a 2 x 4 matrix. I know this

because looking at the numbers going across, there are

2 rows. Looking at the vertical numbers, there are 4

columns.

Difference between types of systems.

 There are three types of systems. These systems include Dependent, Independent, and Inconsistent.

 A dependent system is a system that has infinitely many solutions and they are all numbers.
Example:

 The graphs of the equations are the same.


An Independent system is a system with only one solution.
Example: 











The graphs of the equations are different and they intersect.

An Inconsistent system is a system that has no solution.

These lines are parallel. Because they never touch, there is no solution.