Math 154B
Elementary Algebra
Spring 2012
Study Guide for Exam 4
Exam 4 is scheduled for Thursday, May 3rd. You may use a 3" x 5" note card (both sides) and a
scientific calculator. You are expected to know (or have written on your note card) any formulas
you may need. Think about any rules and procedures you needed to know for homework. For
example: the Pythagorean Theorem, etc...
For Exam 4 you will need to be able to:
1. Simplify square root expressions. For every 2 of the same factors, 1 comes out. 9.1
* positive roots:
* negative roots:
* imaginary roots:
2. Rational and irrational numbers. 9.1
* Perfect square numbers are rational:
1 , 4 , 9 , 16 , 25 ,…
* All other numbers are irrational: 2 , 3 , 5 , 6 , 7 …
3. Simplify square root expressions. 9.2
* Factor numbers down to primes and circle groups of two of the same factors. For every
2 of the same factors, 1 comes out. Leftovers (numbers without partners) stay in.
Multiply all numbers that come out and multiply all numbers that stay in.
* For variables, divide each exponent by 2. The result becomes the exponent on the
variable outside. Obtaining a remainder from the division means one of the variables
7
1
stays inside. Ex: x 7
3
x3 x
2
2
4. Add or subtract square root expressions. You might need to simplify square roots before
they can be combined. 9.3
* All variables (inside and out) and roots have to be exactly the same to add or subtract.
* Just add or subtract coefficients and keep variables and roots exactly the same.
a 2 a
Ex: a
5. Multiply two square roots by using the distributive property or the FOIL method. If
possible, simplify any square roots that appear in the product. 9.3
ab
* Product rule: a b
c)
ab
ac
* Distributive property: a ( b
* FOIL: ( a
b )( c
d)
ac
ad
bc
bd
2
2
x
* Remember: x
x and ( x )
6. Simplify a quotient involving square roots. 9.4
a
b
* Quotient rule:
a
b
7. Rationalize the denominator. Rationalizing means to get rid of the root in the
denominator. You can simplify, then rationalize, or rationalize, and then simplify.
* For 1 term in the denominator, multiply top and bottom by whatever is needed to get rid
of the root.
Ex:
a
b
a
b
b
b
ab
b
* For 2 terms in the denominator, multiply top and bottom by the denominator’s
conjugate. Remember: (1st 2nd )(1st 2nd ) (1st ) 2 (2nd ) 2
x
y x x x y
x y
x
y x
y
8. A square root is completely simplified when…
- No perfect squares or variables with exponents greater than 1 under the root.
x
Ex:
-
2
x, x3 x x , x4
Ex: x , x
No fractions under the root.
Ex:
-
a
b
a
b
b
b
x2 ,...
ab
b
No roots in the denominator.
Ex:
1
b
1
b
b
b
b
b
9. Solving radical equations by… 9.6
* For one radical: get the radical alone on one side of the equal sign, square both sides
to the power of the index, and solve the remaining equation.
Ex:
Solve for x : x a
x
( x )2
a
(a) 2
x
a2
* For two radicals: get each radical to each side of the equal sign, square both sides, and
solve the remaining equation.
Ex:
Solve for x : x
a
x
( x )2
a
( a )2
x
a
10. Solve application problems that involve square roots. 9.7
* These problems involve the Pythagorean Theorem and other formulas involving roots.
__________ ___________ __________
OR

 
 
 
 

(leg)2
Diagonal of solid
d
(other leg)2
(hy potenuse)2
l 2 w2 h2
Practice Problems
The answer to all the problems listed below, even and odd, are in the back of the book. For
those of you who have the Chapter Test Prep Video cd that came with the book, you can use it
to see someone solving each of the problems in the Chapter Tests. If you don't have it, it is
available at the math lab.
Chapter 9 Practice Test
Page 590 #6, 7, 8, 10, 13, 16, 19, 20, 24, 25
Also try:
Page 588 #11, 18, 26, 28, 32, 37, 47, 50, 55, 63, 66, 71
Math 154B
Chapter 9 Exam Review
Name ___________________
1. State whether the root is rational or
irrational.
169
a.
2. Simplify.
125
4. Simplify.
8x 3
6. Simplify.
(3a 4b)2
8. Multiply.
( 4a2b )2 ( 9ab )2
200
b.
3. Simplify.
121
169
5. Simplify.
48x7 y 6 z 5
7. Multiply.
7x
7x
9. Multiply.
3a2b3
24a5b7
10. Simplify.
108
11. Simplify.
75
27
12. Simplify.
y 8a3 y a 2ay3
14. Multiply.
(3 5 2 ) 2
2 9a 5 a 8 a
13. Multiply.
6( 2
8)
15. Multiply.
(2 m
3n)( 2 m 3n)
16. Divide and simplify.
125a 5b 7
5a 3b3
18. Rationalize the denominator.
90 x 2 y 5
2 x3 y 2
19. Rationalize the denominator.
9
a 9
2 xy
7 xy 2
20. Rationalize the denominator.
5
8
17. Divide and rationalize the denominator.
3
21. Solve for m .
m 3
2
22. Solve for y .
2y 3
23. Solve for k .
y 6
3k 1 3
24. Solve for x .
3x
1
4
k
25. Solve for m .
0
3m 3 m 1
26. Find the missing length by using the Pythagorean Theorem, a 2 b 2
26
x
10
c2 .
27. Find the diagonal of a rectangle with length is 15 inches and width 8 inches by using the
Pythagorean Theorem, a 2 b 2 c 2 .
28. A 14-foot ladder is placed 3 feet away from a wall. How far up the wall will the ladder the reach?
29. A square has an area of 784 square meters. Determine the length of each side.
s
A s2
s
30. Find the velocity of a tennis ball dropped from a height, h , of 2 ft as it approaches the ground.
The formula for the velocity, v , in feet per second is v
2gh , where g 32 .
31. Find the distance between the points (-2, -3) and (6, -4) using the distance formula
32. Find the radius of the cone whose area is 60 cube feet using the volume formula, V
3.14 and h 4 .
r 2 h , where
33. The length of a diagonal, d, for a rectangular solid is d
l 2 w2 h2 , where l is the length, w is
the width , and h is the height. Find the length of the diagonal for a rectangular solid if the length
is 3 inches, the width is 4 inches, and the height is 5 inches.
L
,
g
3.14 , L is the length of the pendulum in feet, and g is acceleration due to gravity (32
where
feet per second squared). Find the period of a pendulum if it’s length is15 feet.
34. The period or time, T, it takes (in seconds) for a pendulum to swing back and forth is T
35. Give an example that shows:
a 2 b2
a2
b2 .
2