Merced College; Don Power

 

INTERMEDIATE ALGEBRA STUDY GUIDE -- CH 7 and 8

 

 

7.1       Convert rational exponents to powers and roots, and vice versa; use the results to:

            Evaluate roots;

            Evaluate integers and fractions with rational exponents (positive or negative)

 

            Simplify   −

 

            Part 1:  The sign of the answer is:         

                        a)  Positive       b)  Negative    

 

            Part 2:  The absolute value of the answer is                   

                        a)      b)       c)       27           d)  729

 

 

7.1       Apply properties of exponents to simplify/combine expressions with rational exponents 

 

            Simplify   Hint:  Convert to rational exponents     

            a.   c   b.     c.       d.      e. 

 

 

7.2       Simplify a radical expression containing integers and variables to various powers

                       

 

            Simplify 

            a.        b.        c.        d. 

 

 

7.2       Rationalize a denominator

                        Be able to do the following as a square root, a cube root, or a 4th root:

 

            Simplify         

 

7.3       Simplify expressions and add like radicals

                        for square roots or cube roots

                        up to 2 variables

                       

 

            Simplify and combine like radicals: 

 

            a.       b.          c.    25      d.       e. 75

 

7.3       Rationalize denominators to simplify and add like radicals.

           

            Simplify and subtract:           

 

 

7.4       Multiply and simplify radical expressions

            Distributive law

            FOIL

            Perfect squares

 

            Multiply and simplify: 

 

 

7.4       Divide radical expressions with up to 2 terms by rationalizing the denominator

 

            Rationalize the denominator: 

 

 

7.5       Solve equations with radicals

                        Isolate a radical term

                        Square both sides

                                    If squaring an expression with two terms, remember the middle terms

            Types:

                        Square root requiring squaring of a binomial

                        Cube root

                        Equation with two square roots and a constant

 

            Solve: 

 

            Solve  

 

 

7.5       Apply the Pythagorean theorem

 

            In a right triangle, the hypotenuse is 7 ft, and one leg is 3 ft.        How long is the other leg?

(Give both an exact answer (in radical form, simplified) and an approximation to three decimal places.)

 

 

7.6       Add or subtract two complex numbers:

 

            Subtract:  (3 – 2i) – (4 – i);  write the result in standard form

 

 

7.6       Divide two complex numbers and write the result in the form a + bi

            Rationalize the denominator; remember i2  = -1

           

            Divide:  and write the result in standard form      

 

 

7.6       Solve for both x and y in a complex number equation:

 

            Solve for both x and y:

            6y − 3xi = −12 + 3yi               

 

 

8.1       Solve quadratic equations by factoring

 

            Solve  5x2 = 2x

 

            Solve  4x (4x – 5) = 6

 

 

8.2       Solve quadratic equations by the square root principle

 

            Solve:  x2 = 11 using the square root principle

 

            Solve:  (3x + 2)2 =  −7  using the square root principle

 

 

8.2       Solve quadratic equations by completing the square

 

Solve  x2 – 6x +11=0 by completing the square

 

To solve x2 + 14x = −5 by completing the square, what number must
 be added to both sides of the equation?

 

            a.   2                b.  7                 c.  14               d.  28               e.  49              

 

 

8.3       Solve quadratic equations by the quadratic formula

 

              Solve 5x2 + 2x = 1 by using the quadratic formula:

 

            a.                   b.                           c.                          d.   

 

8.3       Use the discriminant to identify the number and type of solutions:

 

            Matching:  Match the value of the discriminant b2 - 4ac with the number and types of solutions:

 

 

 

Value of discriminant

Number and types of solutions

 

_______

16

a.  One rational solution

 

_______

23

b.  One irrational solution

 

_______

-7

c.  One complex solution

 

_______

0

d.  Two rational solutions

 

 

 

e.  Two irrational solutions

 

 

 

f.  Two complex solutions

 

8.3       Solve a rational equation that results in a quadratic equation when the fraction is cleared.

 

            Solve and give all valid solutions:  

 

Solve:    Give the exact solutions and decimal approximations valid to 2 decimal places

 

 

8.3       Write a quadratic equation that has the given solutions:

 

              Write an equation for which the solutions are x = –5 and x = 2

 

 

8.4       Solve an equation that is quadratic in form

 

            Solve       = 0

 

            Solve  .  Give the sum of the valid solutions.

 

            a.  −2               b.  2                 c.  16               d.  36               e.  52

 

 

8.4       Sketch the graph of a quadratic function.  Give the coordinates of the vertex

 

              For the function g(x) = 3x2 −12x +5,

 

            (1)  Give the coordinates of the vertex                                                                          

            (2)  Does the graph open downward or upward ? (Choose one)

                        (3)  What is the y-intercept?

                        (4)  Sketch the graph

 

 

8.5       Problem involving projectile motion.  Given an equation for projectile motion, either

                        1.         Find the times the object will be a a certain height, or

                        2.         Solve for the time the object will hit the ground., or

                        3.         Find the maximum height (or the time the object will reach the maximum height)

 

If an object is thrown upward from the top of a 60-ft building at a rate of 40 ft/sec, its height above the ground after t seconds is given by the formula h = –16t2 + 40t + 60.  At what two times will the object be at a height of 68 ft?  Pick two of the following choices:

 

a.         .16 sec             b.         .22 sec             c.         .50 sec             d.         2.00 sec

e.         2.28 sec           f.          3.26 sec   

 

 

8.6       Solve the inequality.  Graph the solution set, and give the solution in interval notation.

 

            Solve  (x + 4) (x − 5) ≥ 0

 

            a.  x = −4 or 5       b.  -4 £ x £ 5             c.  x £ -4 or x ³ 5     d.  x ≥ -4 or x ≥ 5

 

 

 

Return to:  Merced College; Don Power               Updated 11/09/07 by Don Power