Aim: How do we multiply and divide rational expressions?
       To multiply or divide a rational expression, you have to factor out any expression that can be factored and then multiply or divide. 

Example
3x-9/x-2*x^2-2x/x-3

1) Factor 3x-9 
3(x-3)/x-2

2) Factor x^2-2x
x(x-2)/x-3

4) Divide x-3/x-3 and x-2/x-2 
*You can cross them out as they both equal to 1

5) Multiply the remaining 3 and x on the numerator 
3x

 Excluded Value:
Find any problem that would make the denominator equal to zero.
x-2=0                           x-3=0

  +2  +2                          +3  +3

x=2                               x=3
x canont be eqaul to 2 or 3

Example
(2b^2-12b/b+5)/(b-6/b+5)

1) Keep the the fraction on the left but change the division sign and fraction 
on the right
(2b^2-12b/b+5)*(b+5/b-6)

2) Factor 2b^2- 12b
2b(b-6)

3) Divide b-6/b-6 and b+5/b+5
* Cross them out because they equal to one

4) This only leaves 2b on the numerator 
2b 
Excluded value:

b+5=0            b-6=0         

  -5  -5              +6  +6

b= 5                b=6

b cannot be equal to 5 or 6.

* Note you also use the denominator you                                                                                                      get after flipping 

How do we simplify expressions with rational exponents?

Aim: How do we simplify expressions with rational exponents?

A rational exponent contains both an integer exponent and a root (the number that must be multiplied by itself  a given times to equal a given value). The root is at the denominator and the integer exponent is at the numerator.  To simplify expressions with rational exponents, you first take the n root of the radicand and then simplify the exponent. 


First example
(16)^5/4
First, you find the 4th root of 16
4√16=2
Then, you multiply 2 by itself 5 times
(2)^5
Now this gives you the answer
32

Second example
(4^1/2)
First, you find the square root of 4
√4=2
Then you get
 4^1
Finally you get  the answer
4

Citation
http://www.mathwords.com/r/rational_exponents.htm

How do we solve radical equations?

Aim: How do we solve radical equations?

To solve radical equation, you first set the radical alone and then square both sides of the equation to get rid of the radical sign. After you get rid of the radical, you can solve it like a regular equation and solve for x. After you get the solutions, you check by plugging it into the original equation to see if the solutions make the equation true. 

For example,

√x-7+5=6

First, subtract 5 from both sides to set the radical alone.

√x-7+5=6

        -5  -5

   √x-7=1

Then, square both sides to get rid of the radical sign.

√x-7^2=1^2

*Squaring gets rid of  the radical sign

x-7=1

Now, add 7 to both sides and solve it like a regular equation.

x-7=1

  +7 +7

 x=8

Then, check by plugging it in the original equation.

√(8)-7+5=6

√1+5=6                    TRUE

1+5=6

6=6

Finally, x=8

How to factor by grouping?

Aim: How to factor by grouping? 

To factor by grouping first separate the equation into two groups and factor out the common factors from each group. Then combine the the common factors that are extracted from each group. 

Example:

x^3 - 3x^2 + 2x - 6 

1) First, separate it into two groups
(x^3 - 3x^2) + (2x - 6)

2) Then factor out the common factors from (x^3 - 3x^2)

The common factor in this group is x^2 and if you take that out      what is left is (x-3)

x^2(x-3)

3) Now factor out the common factors from  (2x - 6)

The common factor in this group is 2 and if you take that out      what is left is (x-3)

2(x-3)

*Note both groups have same expression (x-3) left after taking out the common factor and that ensures the factoring is correct. 

4) Lastly, combine the common factors extracted from each group

Final answer: (x^2+2)(x-3)

* Note you do not write (x-3) twice since is the same. 

What is common factor?

Factors that are common to two or more numbers are said to be common factors.

Citation (image):

- http://misscalculate.blogspot.com/2011/12/factoring-ax2-bx-c.html