10a)   Solve the radical equation

                                                           

     SOLUTION
     To solve the   equation for       , you must isolate one of the square roots.
     It does not matter which one.

      To be specific, isolate    .


                
     so

                

     Square both sides of the equation to get rid of the square root on the left side

                

     Multiply out the right side of the equation by using the identity Square of a Difference

                                                                                  

     You can identity      with       and       with    .

     Remember that if you can not identify the terms in your problem so that your problem
      looks exactly like the identity you want to use, then you can not use the identity.
     You have to try to find another identity that you can try to use to work out your problem.

                
     or

                

     You still have a square root in the equation, so you repeat the process
      and isolate    .

     Subtract       and   22   from both sides of the equation , so

                
     or

                

     To further clean up the equation, divide both sides of the equation by       , so

                

     To get rid of the square root on the right side, again square both sides of the equation

                

     Finally, to get       alone on the right side of the equation, add        to both side of the equation

                
     so

                       is your tentative answer.

     Tentative, because you have to check whether       solves the original equation

                      or not.

     Substitute        into this equation.

     Then, is

                    a true relationship or not?

     The left side is

                

     so you have the relationship

                                     which is true.

     So

                      really is the answer.


     10b)   Solve the radical equation

                                                       

     SOLUTION
     In form this problem is exactly like problem 10a).
     Following the same steps, isolate one of the square roots

                

     Square both sides of the equation to get ride of the square root on the left side of the equation

                

     To work out the square on the right side of the equation, use the identity Square of a Sum

                

     where        and   

     so

                

     or

                

     To isolate the square root remaining on the right side, subtract       and        from both
     sides of the equation

                
     or

                

     To further simplify the equation, divide both side of the equation by       , you get

                

     Now, again, you square both sides of the equation to get rid of the square root on the right side

                

     The left side you work out using the identity Square of a Difference

                

     where       and    

                

     This is a quadratic equation, so put it in standard form. To get alone on the right side,
     on both sides of the equation subtract       and add    8

                
     or

                

     This equation can not be factored easily, so you use the quadratic formula to solve it.
     Remember that the solutions to a quadratic equation in standard form

                

     are given by the quadratic formula

                

     For your equation

                

                
                
                

     Substituting these values into the quadratic formula, you get

                

     or

                


                   

     So the tentative solutions are

                

     You say "tentative", because you must check whether these numbers actually are solutions
      of the original equation

                

     Testing these two numbers, you will find that

                
                
     is a solution of the original equation.



     11)   Solve the rational equation

                                                      

     SOLUTION
     The method for solving rational, or fractional, equations is to multiply the equation
     by the lowest common denominator (LCD) of all the fractions in the equation.
     First, you must factor all the denominators. Here, the denominators are

                    ,       , and   

     so the LCD is

                              .

     You multiply both sides of the equation by the LCD in order to get rid of all the fractions
     in the equation. The factors common in the LCD and in the denominators all cancel,
     so the equation is cleared of all the fractions.

                

     Canceling the common factors, you get

                

     Using the Distributive Law to do the multiplications, you get

                
               

     Combining like terms , writing the highest power of        first, and writing
     the equation in standard form with the    0    on one side, you get

                
               

     Multiply the equation by       to make the leading term positive, you get

                

     To solve this equation of the third degree, you factor the left side by grouping

                
     or

                

     Factoring more, using the identity Difference of Two Squares

                
     you get

                

     The solution is now just a few seconds away!
     Using the Zero Products Principle, you set each linear factor in the product
     equal to solve and solve each linear equation

                    gives   

                     gives   

                     gives   

     Remember, when you solve rational equations, these numbers are tentative solutions.
     You have to check whether they are solutions of the original, rational equation.
     You check by substituting each number back into the original, fractional equation
     to see whether it works or not.
     The original, fractional equation is

                

     If you put        or        , you are dividing by       , which is meaningless,
     an undefined operation.
     So       or        are not solutions of the original equation.

     If you put in       , you get

                

     Is this equation true?
     To find out, you do more arithmetic on the left of the equation


                


     The LCD is       , so writing the fractions on the left side of the equation as
     equivalent fractions with        as their common denominator, you get

                
     or

                

     or

                    which is a true statement,

     because the left side of the equation is now the same as the right side of the equation.

     So

                    is the answer


     

12)   Simplify the complex fraction

                                                            


     SOLUTION
     Using the definition of negative exponents as reciprocals, re-write the fraction

                

     Simplify this complex fraction by multiplying the numerator (the top)
     and the denominator (the bottom) of the complex fraction by the LCD
     of the constituent fractions.
     Here
                

     so, do the multiply

                


     Powers of       and       cancel in the numerator and the denominator, so you get


                


     To see if the fraction can be simplified further, factor the numerator and the denominator.
     Remember the identity for the Sum of Two Cubes

                                                                
                        


     So

                
               

     The        in the numerator and the denominator cancel, so


                                         is the final answer.

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