What is an exponential function ?

The general form of the exponential function can be written :
               f(x) = ab^x
a is the initial value and b is the growth rate.

Let's look at this problem :
      The growth of bacteria in a dish is modeled by the function f(t) = 2^ n . For which value of t is f(t) = 32 ?
                                                                                                             3
The first step is to write out the exponential function .
          32 = 2^ t 
                       3
Next find a common base . To do this you can turn 32 into 2^5 because 2 to the fifth power is 32.

The new function becomes : 2^5 = 2^ t 
                                                          3
Now you solve for x : 5 =  t 
                                         3
You do this by multiplying 3 to every side.
You final answer becomes t = 15.


Now Try It Yourself : 
   A population of rabbits double every 60 days according to the formula P=10(2)^ 1 , where P is the population of rabbits on day t. 
                             60
What is the value of t when the population is 320 ? 

How Do We Solve Exponential Equations ?

Exponential Equations is when a variable is the exponent.
  For Example : 100 = 10^x  
        The missing exponent would be 2 because 10 squared is 100.

Let's try  2^11 = 2^x
Here's a rule to remember : When the Bases are the same, the exponents are equal.
So the missing exponent would be 11.

Lets try something harder.  3^x+4 = 3^7
Since the bases are the same, set up an equation with the exponents.
It looks like this : x+4=7
 You solve regularly for x, which becomes x=3.

Lets try one more problem . 2^x-3 = 4
First you have to create common bases. You can do this by turning 4 into 2^2.
Now the exponential equation becomes 2^x-3 = 2^2
You set up an equation : x-3 = 2, which becomes x=5.

Now try it Yourself : 2^3x+7 = 2^5x-1

How Do We Simplify Complex Fractions ?

Let's get       2       in simplest form
                    x     
                    1
                   2x

  The first step is to rewrite the problem as a division problem, so it would look like this :
                2    ÷   1 
                x        2x

  The next step is to divide as you would any fraction. Keep the first fraction, change the sign from division to multiplication and flip the second fraction.
It should look like this :

               2   *   2x 
               x         1

The next step is to multiply straight across.

                     4x  
                     1x
The final and last step is to cross out anything the numerator and denominator have in common which is X .

  Final Answer :    4   -------> 4 
                            1


Now Try It Yourself : 
                6  
               9x       
                5 
               3x

How Do We Add And Subtract Rational Expressions ?

To add and subtract rational expression, you have to first find a common denominator.

                                                    X-1                X-5        <------ Numerator
                                                  X +6                X+6       <------ Denominator
Lets take :

              u-v               6u -3v
              8v       +          8v 
 The first step in solving this is finding a common denominator. Since this rational expression already is in a common denominator, you don't have to worry about it.

Next you add or subtract the terms, according to what the question ask you. In this case, You add by adding like terms.

          So .....                  u + 6u = 7u   and  -1v-3v = -4v 
                Together the numerator is  7u-4v
                               the denominator is 8v.


NOW TRY THIS : 

                     7p+9z     -     3p+8z
                         9g                 9g

How Do We Solve Fractional Exponent Equations ?

To solve fractional rational exponents you have to get the variable and exponent alone before first.

 Let's use :  x^4/3 - 6 = 10 .

The first step is to add 6 to both sides to get x^4/3 by itself .
        The equation now looks like : x^4/3 = 16
The next step is to find the reciprocal of the exponent ( 4/3 ) and multiply it to both sides. This gets x by its self.

       Visual :  (x^4/3) (3/4) = 16 (3/4)
                          x = 16 ^ (3/4)
Next you have break up the exponent. You do this by putting 16 ^ (1/4) first. Then squaring that by 3.

           Visual :                               16^ (1/4) = 2.
                                            2^3 = 8 
                                 Your final answer is 8 . 

Now Try It Yourself :
                                        8^(2/3)

* Remember : (1/2) = square roots 
                        (1/3) - cubed root 
                                           
                   

How Do We Solve Radical Equations ?

Lets use the example :
The first step in solving radical equations is making sure the radical is by itself.

Let's take √x-7 + 5 = 6 , you would have to subtract 5 from each side to get the radical by itself. 
The next step is to square each side of the equation. Do not square terms. 

   Here is an example : 

               CORECT : √x-7^2 =1^2
        
                              x-7 =1
The next step is to solve the equation by solving for X. You can do that by getting X alone. In this case it would look like : 

                                     x-7 =  1
                           +7    +7
                              x = 8
The last and final step is to plug the value of X back into the equation to see if it works.


                          √ 8-7 + 5 = 6 ----------> 8-7 is 1

                   √1+ 5  = 6  -----> The square root of 1 is 1.

                           6 = 6  ----------> 1+5 =6

                    The solution X = 8 is true . 



Now try it yourself  :


                             √x-3 = 5

How Do We Factor By Grouping ?

To factor by grouping, you have to use 5 easy and simple steps.
     Let's use the polynomial : 8x^2-10x-3 
The first step is to find the Master Product. The master product is multiplying the first number by the last number. So in this case you would multiply 8 by -3 and get -24.
The next step if to find what multiplies to give you -24 and adds together to give you -10.
                                    The two numbers become -12 and 2.
Next you replace the -10x with the two new factors you found ( -12 and 2 )
              Your polynomial should now look like this : 8x^2-12x+2x-3
 Now its time to group the the terms in two pairs. Group the first two terms and the last two terms
       You have to factor each pair by finding the greatest common factor.
           So you can pull out 4x from 8x^2 -12 . The first two terms now look like :
                                                 4x(2x-3) 
         You can pull out 1 from 2x -3 because there's nothing else that goes into both 2 and 3.
  Both terms together now looks like this :  4x(2x-3) + 1(2x-3)
      The final and last step is to factor out the shared binomial, which is (2x-3) and combine the term in front of the parenthesis.
                       Your final answer is (4x+1) (2x-3) .

Now try it is your self  :

                                              6x^2 + 19x + 10