Math 120 Exam 3 Study Guide Solutions

The study guide does not look exactly like the exam but it will help you to focus your study efforts.

Here is part of the list of items under “How to Succeed in Math 120” that is on my syllabus.

Put aside time to study for this class every day. One suggestion is to rewrite your notes after each

class.

There is no substitute for writing out problems. You literally should do 250 or more problems for

each unit of the course.

There are usually examples completely worked out in the book for the types of problems assigned for

you to utilize before attempting homework.

The back of the book has answers for the ‘odd numbered’ problems. You should always do some of

these before attempting the required problems to help you know if you are on the right track.

Typical Exam Directions:

Use pencil. Simplify all solutions. Show your work. Clearly identify your final solutions.

No calculators. Do not leave negative exponents in your final solutions unless you are required to.

Your Notes combined with the required problems will give you the best idea of what types of problems

you should study to prepare for the exam.

The book has plenty of problems of the following type to use as practice.

Simplifying radicals: Sections 7.1, 7.3, 7.4, and 7.5

Solving radical equations: 7.6

Complex numbers: 7.8

Here are some additional problems to use as practice with special attention paid to functions.

1. Simplify each radical expression. (Assume that all variables are nonnegative.) Some solutions may

require the use of ‘i’ notation.

a) 1044x 52 11x c) 194 y 4 34y y e) 3 2456m = 8 32 7m

b) 3 64 = 4 d) 64 =8i f) 8

8 15 15

an even root always gives positive answer

2. Rewrite each exponent expression in radical form.

a) 3

5 5 3x x b) 1

2b b

3. Rewrite each radical expression in exponent form.

a) 1

26 6 b) 511511m m

4. Evaluate each expression using exponent properties. The solution should be a number with no

exponents or radicals.

a) 1264 8 b)

33 44 4 381 3 3

1

27

5. Simplify each expression using exponent properties. Do not use radicals. Do not leave negative

exponents in final solutions.

a) 5 1

34x x

b)

54

13

x

x

c) 1

5 34x

(Add exponents.) (Subtract exponents.) (Multiply exponents. Rewrite w/out negative.)

15 4

12 12x x

1512

412

x

x

512x

11

12x 19

12x 5

12

1

x

6. Write each of the following expressions in the form: px where ‘p’ is a real number.

a) 2

6

x

x

4x

b) x x 1

2x x 3

2x

c) 3 x

x

13x

x

23x

7.

a) If one leg of a right triangle measures 4 cm and the hypotenuse measures 10 cm, how long is the other leg?

Let ‘a’ and ‘b’ be the length of the legs and ‘c’ be the length of the hypotenuse.

2 2 2a b c Then: 2 2 24 10b

2 84b The other leg measures 2 21 cm

84b 4 21 2 21

b) If one leg of a right triangle is 5 and the hypotenuse measure is x, what is the length of the other leg?

2 2 2a b c Then:

2 2 25 b x

2 2 25b x

2 25b x This is the length of the other leg of the right triangle.

8. a) What is the distance between the points 1,4 and 2,6 ?

2 2

2 1 2 1d x x y y 2 2

2 1 6 4 9 4 13

b) What are the coordinates of the midpoint of the segment that joins 1,4 and 2,6 ?

Midpoint 1 2 1 2,2 2

x x y y

1 2 4 6

,2 2

1

,52

9. Determine if the given expression is rational, irrational, or imaginary.

4 2i 4 =2 3 4 3 4

Imaginary rational irrational (This is still a real number)

Can be written as 21

10. Solve each radical equation. Check your solutions.

a) 1 5x x c) 5 5y y

2 2

1 5x x 2 2

5 5y y

21 10 25x x x 5 25 5 5y y y y

20 11 24x x 5 25 10 y

0 3 8x x 20 10 y

3,8x (Check both solutions.) 2 y

Only 8x works! 4y (Check the solution. It works)

b) 3 4 3 7 5x

3 4 3 2x

4 3 8x

4 5x

54x

11. List the real part and the imaginary part of each complex number.

4 7i 6 3

11

i

Real part = 4 Real Part = 611

Imaginary part = 7i Imaginary part = 311

i

12. Answer and simplify each question based on the given functions. Rationalize all denominators.

( )f x x 7x 3( )p x x

a) Domain of p(x). Use interval notation. (x is any real number) ,

b) (8) (18) 8 18f f 2 2 3 2 5 2

c) 3

1 1

(2) 2p

3 3

3 3 3

1 4 4

2 4 8

3 4

2

d)

1

2f

1

2

1 2

2 2

2

2

e) Domain of g(x). Use interval notation. (solve 4 3 0x ) 43,

e) Find all x such that ( ) 5g x . 4 3 5x

Square both sides and then solve.

Solution. 7x

f) (2 ) (2)

h

f h f

2 2

h

h

2

22

2

22

h

h

h

h

2 2

2 2

h h

h

2 2h h

h

2 2h

3 1y x (down 1)

13. Write the function for each graph.

4y x (left 4)

1

12

yx

(right 2, up 1) 3

34

y x (slope -3/4, b = 3)

15. Graph each of the following functions using the methods taught in class.

a) ( ) 2f x x d) 2

( ) 3 2f x x

left 2 units upside down, right 3, up 2

b) ( )f x x e) ( ) 3 2f x x

upside down line with slope 3/1 and y-intercept -2

c) ( ) 1 3f x x f) 1

( )4

f xx

right 1 unit and down 3 units left 4 units

Page 1 of 4

Math 120 Ch 7 Practice Solutions Use pencil. Simplify all solutions. Show your work. Clearly identify your final solutions. No calculators. Do not leave negative exponents in your final solutions. 1. Simplify each radical expression. (assume that all variables are nonnegative)

a) 1436b = 76b f) 194 y 4 34y y k) 3 15y y 3 5y

b) 144m 12 m g) 193 y 6 3y y l) 718y

33 2y y

c) 3 64 = 4 h) 3 11125a

33 25a a m) 13 26 = 13 2

d) 64 = 8i i) 3 2456m = 8 32 7m n)

3 23 9 3x x = 3x

e) 1044x 52 11x j) 20 = 2 5i

2. Simplify each radical expression. Rationalize all denominators.

a) 2 5 2x x = 5 2 2x x

b) 2

5 2 = 25 5 2 5 2 2 = 27 10 2

c) 2

5

2 5 2 5

55 5

d) 4

5 2

4 5 2

5 2 5 2

4 5 2

25 2

20 4 2

23

e) 32

5

9x=

3 3

33 2

5 3

39

x

xx

3 15

3

x

x

f) 87 76 2x y x xy 7 76 2x xy x xy 74x xy

3. Rewrite each exponent expression in radical form.

a) 3

5 5 3x x b) 1

2b b

4. Rewrite each radical expression in exponent form.

a) 1

26 6 b) 511511m m

5. Evaluate each expression using exponent properties. The solution should be a number with no exponents or radicals.

a) 1

264 8 b) 33 44 4 381 3 3

1

27

Page 2 of 4

6. Write the expression as a single radical. (Hint: First convert to exponents.) 5 24 t t

21

545 24 t t t t 5 8

20 20t t 13

20t 20 13t

7. Determine if the given expression is rational, irrational, or complex (nonreal).

4 4 3 4 Complex (nonreal) OR Imaginary rational irrational 8. Simplify each complex number expression.

a) 5 3 3 4 8i i i

b) 5 3 3 4 2 7i i i

c) 2 7 1 4 3 3i i i

d) 2

4 4 4 4

7 7 7 7

i i i

i i i i

4

7

i

e) 5 1 4i i = 25 20i i 20 5i

f) 24 7 2 3 8 12 14 21i i i i i 29 2i

g) 9 144 3 12i i 15i

h)

5 3 3 45 3

3 4 3 4 3 4

i ii

i i i

2

2

15 20 9 12

9 16

i i i

i

3 29

25

i

3 29

25 25i

9. Solve each radical equation. Show your work. Check your solutions.

a) 5 8x b) 5 2x c) 3 5 2x

5 64x no solution 5 8x 59x 13x 10. Solve each radical equation. Check your solutions.

a) 1 5x x c) 5 5y y

2 2

1 5x x 2 2

5 5y y

21 10 25x x x 5 25 5 5y y y y

20 11 24x x 5 25 10 y

0 3 8x x 20 10 y

3,8x (Check both solutions.) 2 y

Only 8x works! 4y (Check the solution. It works)

Page 3 of 4

b) 3 4 3 7 5x

3 4 3 2x

4 3 8x

4 5x

54x

11. Answer and simplify each question based on the given functions. Rationalize all denominators.

( )f x x ( ) 4 3g x x 3( )p x x

a) Domain of p(x). Use interval notation. (x is any real number) ,

b) Domain of g(x). Use interval notation. (solve 4 3 0x ) 43,

c) (8) (18) 8 18f f 2 2 3 2 5 2

d) 3

1 1

(2) 2p

3 3

3 3 3

1 4 4

2 4 8

3 4

2

e) (2 ) (2)

h

f h f (for fun )

12. Graph each of the following functions using the methods taught in class.

a) ( ) 2f x x d) 2

( ) 3 2f x x

b) ( )f x x e) ( ) 3 2f x x

Page 4 of 4

c) ( ) 1 3f x x f) 1

( )4

f xx

13. List the real part and the imaginary part of each complex number.

√

Real part = 4 Real Part = 6 38 4

Imaginary part = Imaginary part = √

√

14. Find the distance between the points ( ) and ( ).

√( ( )) ( )

√( ) ( )

√

√ 15. Use Pythagorean’s Theorem to solve for the missing side in each right triangle. (Sec 7.7)

a) b) √ √

(√ ) ( √ )

( )

√ √ √ √

| | √ √ | | √ √

√ √