The University of Texas at El Paso
Tutoring and Learning Center
ACCUPLACER
MATH 0300
http://www.academics.utep.edu/tlc
MATH 0300
Page
Fractions
2
Fractions – Exercises
7
Fractions – Answers to Exercises
8
Decimal Numbers
10
Decimal Numbers – Exercises
12
Decimal Numbers – Answers to Exercises
13
Percents
14
Percents – Exercises
16
Percents – Answers to Exercises
17
Order of Operations
18
Order of Operations – Exercises
19
Order of Operations – Answers to Exercises
20
Real Numbers, the Number Line, Order Relations and Absolute Value
21
Real Numbers, the Number Line, Order Relations and Abs. Value – Exercises
23
Real Num., the Num. Line, Order Rel. and Abs. Value – Ans. to Exercises
24
Scientific Notation
25
Scientific Notation – Exercises
26
Scientific Notation – Answers to Exercises
27
Exponents
28
Radicals
31
Laws of Exponents and Radicals
33
Exponents and Radicals – Exercises
34
Exponents and Radicals – Answer to Exercises
35
FRACTIONS
A fraction is defined as a ratio of two numbers, where the number at the bottom cannot be equal to
zero.
a
where b ≠ 0
b
In a fraction the number at the top is called the numerator, and the number at the bottom is called the
denominator.
a
numerator
=
b denominator
There are two different kinds of fractions, proper and improper. In a proper fraction the numerator
(the number at the top) is less than the denominator (the number at the bottom). In an improper
fraction, the numerator is greater than the denominator.
numerator < denominator
Example:
2
3
Improper fraction: numerator > denominator
Example:
5
4
Proper fraction:
When you have an improper fraction, it is not always right or recommended to leave it as your final
answer. It is always best to change an improper fraction to a mixed number. A mixed number is a
whole number and a fraction together.
Mixed number:
C
a
a
is a fraction
→ C is a whole number and
b
b
To change an improper fraction to a mixed number we need to divide the numerator by the
denominator.
2
Let’s take our previous example of an improper fraction to make it into a mixed number:
Denominator
Example:
5
4
→
Whole number
1
4 5
−4
1
→
1
1
4
Numerator
When working with a fraction that contains large-value-numbers, it is always best to reduce the
fraction to an equivalent fraction. An equivalent fraction is a fraction that has the same value but
contains different numbers. To find an equivalent fraction, simply multiply the numerator and
denominator by the same number.
Example:
3 3 3 9
= ⋅ =
5 5 3 15
then
3 9
=
5 15
In the case of reducing a fraction, or simplifying the fraction as you may also call it, we need to divide
the numerator and denominator by the same number.
Example:
10 10 ÷ 5 2
=
=
25 25 ÷ 5 5
therefore
10 2
=
25 5
We have now found a fraction that is of the same value as the original one, but contains smaller digits
and is easier to work with.
Now let’s get into the basic operations of fractions.
1. Addition or subtraction of fractions with the same denominator.
When you need to add or subtract fractions that contain the same denominator, all you need to do is
to add or subtract the numerators, depending on the problem, and keep the same denominator.
a c a±c
± =
b b
b
Example:
3
Addition:
2 1 2 +1 3
+ =
=
5 5
5
5
Subtraction:
5 3 5−3 2
− =
=
9 9
9
9
2. Addition or subtraction of fractions with different denominators.
To add or subtract fractions with different denominators, we must first find a common number
between the two denominators making the problem easier to solve. This number is called least
common denominator (LCD). Using the LCD makes the same denominator for all the fractions
and the operation will be easier to perform.
Suppose we have the following problem:
3 2
+
5 9
To find the LCD, we must think of a common multiple for both denominators. Our denominators
are 5 and 9 their multiples are as follow.
Multiples of 5:
Multiples of 9:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50 …
9, 18, 27, 36, 45, 54, 63, 72, 81, 90 …
As you can see our LCD is 45 since it is the first multiple that both numbers have in common.
Now that we have found our LCD we can continue on with the operation. Rewrite both fractions to
equivalent fractions with 45 as their denominator:
27 10
3 2
+
→
+
5 9
45 45
We now have an addition of fractions with the same denominator and are now able to perform the
operation:
27 10 37
+
=
45 45 45
3. Multiplication of fractions.
In multiplication of fractions, multiply numerators with numerators and denominators with
denominators.
2 4 2× 4 8
× =
=
5 7 5 × 7 35
4
4. Division of fractions.
To divide fractions we first need to invert the second fraction and then perform a multiplication of
fractions.
1 3
1 2
÷
→ ×
=
4 2
4 3
1× 3 3
=
4× 2 8
Multiply both fractions
Invert the second fraction
5. Operations with whole numbers.
When dealing with whole numbers, we must first convert them into fractions. To do this simply put
a 1 as the denominator and treat it as a fraction.
2+
3
2 3
8 3
11
=
+
=
+
=
4
1 4
4 4
4
Put a 1 as the
denominator
Obtained the LCD and
perform the operation
6. Changing fractions to decimals.
To change fraction into decimals we simply need to divide the numerator by the denominator.
5
4
→
1.25
4 5
−4
10
−8
20
− 20
0
Therefore:
5
= 1.25
4
7. Fraction proportions.
Proportions are equivalent fractions that are missing a number, either the numerator or the
denominator. We use cross multiplication to find the missing number.
Example:
5
1 x
=
4 3
→
(1)(3) = (x )(4)
→
3 = 4x
→
x=
3
4
8. Word problems containing fractions.
When working with word problems that involve fractions, follow the procedures described
previously to solve the problem.
Example: A rectangular piece of material 3 feet wide by 12
1
feet long is cut into five equal
2
strips. Find the length of each strip.
→ Find the length of each strip by cutting the piece length wise, so divide 12
1
by 5:
2
5
1
1
25 5 25 1 25
12 ÷ 5 =
÷ =
× =
=2 =2
10
2
2
2 1 2 5 10
→ Therefore, each piece will be 2
1
feet.
2
6
FRACTIONS – EXERCISES
Reduce:
1. −
35
70
2.
77
121
3.
16
56
4.
−9
15
5.
15
27
6.
40
35
7.
18
24
8.
48
54
9.
19
76
10.
20
−75
Find x:
11.
2 9
=
12 x
12.
3 x
=
4 12
13.
15 x
=
10 30
14.
1 3
=
12 x
15.
55 22
=
5
x
16.
7 14
=
x 3
Perform the indicated operations and simplify your answer:
17.
8 9
+
3 4
18.
11 5
−
3 6
19.
1 2
−
4 9
20.
1 3 1
+ +
6 8 3
21.
13 2
+
17 11
22.
5
+7
3
23.
1 3 1
+ +
3 7 6
24.
5 6
−
3 7
25.
9−
26.
2 1
⋅
3 6
27.
7÷
1
9
28.
1
÷4
8
7
3
8
FRACTIONS - ANSWERS TO EXERCISES
35
35 ÷ 35
1
=−
=−
70
70 ÷ 35
2
2.
77
77 ÷ 11 7
=
=
121 121 ÷ 11 11
16 16 ÷ 8 2
=
=
56 56 ÷ 8 7
4.
9
9÷3
3
−9
=− =−
=−
15
15
15 ÷ 3
5
5.
15 15 ÷ 3 5
=
=
27 27 ÷ 3 9
6.
40 40 ÷ 5 8
=
=
35 35 ÷ 5 7
7.
18 18 ÷ 2 9
=
=
24 24 ÷ 2 12
8.
48 48 ÷ 6 8
=
=
54 54 ÷ 6 9
1.
−
3.
9.
11.
13.
15.
19 19 ÷ 19 1
=
=
76 76 ÷ 19 4
2 9
=
12 x
2 x = 108
x = 54
15 x
=
10 30
10 x = 450
x = 45
55 22
=
5
x
55 x = 110
10.
12.
8 9 32 27 59
+ =
+
=
3 4 12 12 12
3 x
=
4 12
4 x = 36
x=9
14.
1 3
=
12 x
x = 36
16.
7 14
=
x 3
14 x = 21
3
x=
2
18.
11 5 22 5 17
− =
− =
3 6 6 6 6
x=2
17.
20
20
20 ÷ 5
4
=−
=−
=−
−75
75
75 ÷ 5
15
8
19.
1 2 9
8
1
− =
−
=
4 9 36 36 36
20.
8
21 7
1 3 1 4
9
+ + =
+
+
=
=
6 8 3 24 24 24 24 8
21.
13 2 143 34 177
+ =
+
=
17 11 187 187 187
22.
5
5 21 26
+7 = + =
3
3 3
3
23.
1 3 1 14 18 7
39
+ + =
+
+
=
3 7 6 42 42 42 42
24.
5 6 35 18 17
− =
− =
3 7 21 21 21
25.
9−
5
3 72 3 69
=
− =
=8
8
8 8 8 8
26.
2 1 2 1
⋅ =
=
3 6 18 9
27.
7÷
1 7 1 7 9
= ÷ = × = 63
9 1 9 1 1
28.
1
1 4 1 1 1
÷4 = ÷ = × =
8
8 1 8 4 32
9
DECIMAL NUMBERS
1. Addition or subtraction of decimal numbers.
When adding or subtracting decimal numbers, the first step is to align the decimal point.
Example:
Add 7.01 and 6.3
7.01
+ 6.30 ← Add a 0 to align all numbers
13.31
Subtract 29.01 and 17.43
29.01
− 17.43
11.58
2. Multiplication of decimal numbers.
When multiplying decimals there is no need to align the point. However, the decimal point in
the final answer will be placed according to the total number of places to the right of the
decimal point.
Example:
Multiply 1.25 and 3.2
1.25
× 3.2
250
375
4.000
← 2 digits to the right of the decimal point
← 1 digit to the right of the decimal point
← Total of 3 digits to the right of the decimal point
← Count three digits from right to left to place the decimal point
3. Division of decimal numbers.
When dividing decimals, the divisor must contain no decimal places. If it does, move the
decimal place to the right until it becomes a whole number. The number of places you move the
decimal point on the divisor will indicate the number of places you need to move your point in
the dividend.
10
Example:
Divide 117.525 by 2.5
47.01
2.5. 117.5.25
− 100
Move the decimal point
one place to the right; this
will be your new decimal
place.
175
− 175
0 25
− 25
0
4. Change decimals into fractions.
To change decimals into fractions the first step is to write the decimal number with no decimal
point. This will be the numerator. The denominator will be 10, 100, 1000, etc, depending on the
numerator. If it is a single digit number it will be 10, a double digit number will get 100 as the
denominator, and so on.
Example:
Change 0.41 into a fraction.
Step 1: Write the decimal number with no decimal point, do not write the
41 will be the numerator
zero: 0.41 → 41
Step 2: The denominator in this case will be 100, since 41 is a double
41
digit number:
100
41
Therefore:
0.41 =
100
5. Word problems containing decimals.
To solve word problems that contain decimals, follow the procedures described above
depending on the type of problem.
Example:
If you earn $526.35 and $64.52 is taken out on taxes, what is your total after-tax
earnings?
Subtract the amount of money taken out from your total amount to obtain the
total after-tax earnings:
526.35
− 64.52
461.83 ← Amount after-tax.
11
DECIMAL NUMBERS - EXERCISES
Perform the indicated operations.
1.
76.19 + 19.2
2.
56.4 + 20.6
3.
197.64 – 129.37
4.
20.1 – 18.367
5.
2. 03 × 6. 3
6.
17.88 × 2.19
7.
6.81 × .04
8.
195.6 ÷ 16.3
9.
6. 81÷. 03
10. 17.88 ÷ 2.19
11. If one pound of apples costs $0.79, how much would 2 1/2 pounds cost?
12. If a 14 1/2 pound watermelon costs $2.99, how much does the watermelon cost per pound?
13. If you earn $35.25 on Saturday and $31.75 on Sunday, what is your total pay for the weekend?
14. If you earn $236.05 and $39.73 is taken out for taxes, what is your total after-tax earnings?
12
DECIMAL NUMBERS – ANSWERS TO EXERCISES
1.
76.19
+ 19.20
95.39
2.03
× 6.3
5.
609
1218
12.789
227
.03. 6.81
6
8
9.
6
21
21
0
.206
14.5. 2.9.900
2 90
12.
900
870
30
13
56.4
2. + 20.6
77.0
17.88
× 2.19
16092
6.
1788
3576
39.1572
3.
197.64
− 129.37
68.27
6.81
×
.04
7.
.2724
8.16
2.19. 17.88.00
17 52
360
10.
219
1410
1314
96
$35.25
+
$31.75
13.
$67.00
20.100
4. − 18.367
1.733
12.
16.3. 195.6.
163
8.
326
326
0
$0.79
× 2.5
11.
395
158
$1.975
14.
$236.05
− $39.73
$196.32
PERCENTS
Percent means per cent, or per every hundred. In other words, percent means how much of the whole
amount you have. A percentage can also be represented as a fraction whose denominator is 100.
Examples:
46% means
46
or 46 out of 100.
100
10% means
10
1
or 10 out of 100. It can also be reduced to .
100
10
1. Convert a percent to a fraction.
To convert a percentage into a fraction, divide the percentage number by 100, then reduce if
necessary.
Examples:
32%
→
32
8
=
100 25
25%
→
25
5
=
100 20
2. Convert a percent to a decimal.
To convert a percentage into a decimal shift the decimal point in the percentage two places to the
left.
Examples:
→
7%
0.07
In this case, there is no decimal place shown in the percentage, however, it
exists at the end of the number. Fill the required spaces with a 0.
6.38% →
0.0638
3. Convert a decimal to a percent.
To change a decimal into a percentage shift the decimal point two places to the right.
Examples:
→
36%
0.025 →
2.5%
0.36
14
4. Convert a fraction to a percent.
To convert a fraction into a percentage, first divide the numerator by the denominator and then
change the decimal into a percentage.
Examples:
3
= 0.6 = 60%
5
7
= 0.875 = 87.5%
8
5. Problems involving percents.
When dealing with problems involving percentages, remember that “of” means multiplication and
“is” means equals.
Examples:
What is 15% of 260?
First convert the percent to decimal:
15% = 0.15
Then convert the words into mathematical terms:
or
? = 0.15 × 260
( what ) ( is )(15% ) ( of ) 260
y
= .15
x 260
Finally, solve the problem:
0.15 × 260 = 39
Answer:
15% of 260 is 39.
50 is what percent of 940?
x
In this case the percentage is the unknown. Remember, first convert the words
into mathematical terms:
50 ( is )( what percent )( of ) 940
or
50 = ? × 940
50 =
y
× 940
Solve the problem:
15
50 = ? × 940
50
?=
= 0.053 or 5.3%
940
PERCENTS – EXERCISES
1.
15% of 750 is what number?
2.
85% of what number is 255?
3.
What is 18% of 350?
4.
75 is what percent of 300?
5.
What percent of 450 is 250?
6.
What number is 20% of 45?
7.
195 is 35% of what number?
8.
150 is what percent of 500?
9.
What is 6.5% of 45?
10. 10.5% of what number is 21?
16
PERCENTS – ANSWERS TO EXERCISES
1.
0.15 ⋅ 750 = x
112.5 = x
0.85 ⋅ y = 255
2. y = 255 ÷ 0.85
y = 300
y
⋅ 300
100
75 ⋅ 100
4.
=y
300
25 = y
75 =
3.
x = 0.18 ⋅ 350
x = 63
y
⋅ 450 = 250
100
250 ⋅ 100
5. y =
450
y = 55.56
6.
x = 0.20 ⋅ 45
x=9
195 = 0.35 ⋅ y
195
7.
=y
0.35
y = 557.14
y
⋅ 500
100
150 ⋅100
8.
=y
500
y = 30
x = 0.065 ⋅ 45
x = 2.925
0.105 ⋅ y = 21
21
10. y =
0.105
y = 200
9.
17
150
=
ORDER OF OPERATIONS
When dealing with a problem that contains different types of operations there are a series of steps that
need to be followed.
1) Do the operations inside of parentheses. Then eliminate.
2) If there are any exponents present, simplify the numbers to eliminate of the exponents.
3) Next, do all multiplications and divisions present from left to right.
4) Finally, do all addition and subtractions from left to right.
Examples:
1. Solve:
2. Solve:
3. Solve:
16 × 4 − 2 3 + (15 ÷ 5)
= 16 × 4 − 2 3 + 3 ← First thing is to eliminate the parentheses, 15 ÷ 5 = 3
← Carry out the exponential operation the exponent, 2 3 = 8
= 16 × 4 − 8 + 3
← Do the multiplication, 16 × 4 = 64
= 64 − 8 + 3
← Do additions and subtractions from left to right
= 56 + 3
= 59
11 + 9(7 − 2 )
33 − 27 ÷ 3
11 + 9(5)
=
27 − 27 ÷ 3
11 + 45
=
27 − 9
56
=
18
2
1
=3 =3
18
9
116 − 9 ÷ 3
= 116 − 3
= 113
18
ORDER OF OPERATIONS – EXERCISES
Perform the indicated operations:
1.
7 − 3+ 6× 2 + 9
2. 117 + 8 − 10 × 3 ÷ 6
3.
2 12 7
+  + 
3 45 3
4.
4  8 10 
÷ × 
5  25 16 
2
5
3
5.-   −
 4  12
2
5  5 2
6.-   ÷  + 
 6   12 3 
7.-
7 2 5
÷ + 
12  3 9 
8.-
11  3 
7
−  +
16  4  12
9.-
3  11 7  5
× −  +
4  12 8  16
2
3
9 2 2
10.- ×   +
10  3  3
19
ORDER OF OPERATIONS – ANSWERS TO EXERCISES
1. 7 − 3 + 6 × 2 + 9 = 7 − 3 + 12 + 9 = 25
2. 117 + 8 − 10 × 3 ÷ 6 = 17 + 8 − 30 ÷ 6 = 117 + 8 − 5 = 125 − 5 = 120
3.
2 1  2 7  2 1  6 35  2 1  41 
+  + = +  + = +  
5 4  5 3  5 4  15 15  5 4  15 
=
4.
2 41 24 41 65 13
+
=
+
=
=
5 60 60 60 60 12
4  8 10  4 80
4 400
÷ ×  = ÷
= ⋅
=4
5  25 16  5 400 5 80
2
5  3  3  5
9
5 27 20 7
3
5.   −
=
−
=
−
=
=    −
12  4  4  12 16 12 48 48 48
4
2
8   25   13  25 12 300 25
 5   5 2   5  5   5
6.   ÷  +  =    ÷  +  =   ÷   =
× =
=
 6   12 3   6  6   12 12   36   12  36 13 468 39
7.
7  2 5  7  6 5  7 11 7 9
63 21
÷ = × =
=
÷ +  =
÷ +  =
12  3 9  12  9 9  12 9 12 11 132 44
2
11  3 
7 11  3  3  7 11 9
7 33 27 28 34 17
8.
=
− +
=
−
+
=
=
=
−    +
−  +
16  4  12 16  4  4  12 16 16 12 48 48 48 48 24
9.
3  11 7  5 3  22 21  5 3 1
5
3
5
3 30 33 11
= ×
+
=
+
=
+
=
=
= × −  +
× −  +
4  12 8  16 4  24 24  16 4 24 16 96 16 96 96 96 32
2
10.
9 2
2 9  2  2  2 9 4 2 36 2 36 60 96 16
+ =
+
=
=
×   + = ×    + = × + =
10  3 
3 10  3  3  3 10 9 3 90 3 90 90 90 15
20
REAL NUMBERS, THE NUMBER LINE, ORDER RELATIONS, AND
ABSOLUTE VALUE
When talking about numbers, there are different types of categories in which digits can be classified.
The graphical representation of numbers is called the number line.
− 5 − 4 − 3 − 2 −1 0
1
2
3
4
5
Integers are whole numbers; that is, numbers that contain no decimal point or numbers that are not in
fraction form.
Example:
... , − 4, − 3, − 2, − 1, 0, 1, 2, 3, ...
Integers are classified as positive or negative. Positive integers are numbers that contain no negative
sign. These numbers are placed to the right of zero on the number line. Negative integers are numbers
that contain a negative sign; these are placed to the left of zero on the number line. Zero is neither
positive or negative.
Examples:
Positive integers:
1, 2, 3, 4, 5, …
Negative integers:
− 1, − 2, − 3, − 4, − 5, ...
Even integers are those integers which can be evenly divided by 2. Those integers which cannot be
evenly divided by 2 are called odd integers.
Examples:
Even integers:
2, 4, 8, 10, 12, − 24
Odd integers:
3, 7, 15, 21, 25, − 27
Prime numbers are integers that are greater than 1, but that are not divisible by any integer other than
themselves and 1.
Examples:
Prime integers:
2, 5, 7, 17, 19, 23, 29
When numbers are presented in the form of a fraction they are called rational numbers. When a
number cannot be represented in fraction form this number is called an irrational number.
Examples:
Rational numbers:
1 2 3 3
, , ,
2 3 4 8
Irrational numbers:
21
π , 2, 3
When comparing any two real numbers, refer to the number line to see which number is greater or
smaller. The number to the left is always the smaller number; while the number to the right is always
the greater number of the two. This comparison is called order relations.
Examples:
1 > −1
8>0
3<9
− 15 < 15
− 2 > −6
− 5 < −4
If comparing two fractions, it is always easier to convert the fraction into a decimal and then compare.
Decimals are easier to compare than fractions.
Examples:
1 5
<
2 6
→
0.50 < 0.833
3 3
>
8 4
→
0.375 > 0.75
The absolute value of a number is always the positive value of that number. Since the absolute value
is the distance of a number from the origin, and since distances are always positive, the absolute value
is always a positive value.
Examples:
3 =3
−
3 3
=
4 4
−3 = 3
− 0.25 = 0.25
Reciprocals
The reciprocal of a number is the number to which the first number is multiplied against and the yield
is one.
For example,
1
5⋅ =
1
5
−1
−4 ⋅ =1
4
1
x⋅ =
1
x
The reciprocal of 5 is 1/5
The reciprocal of -4 is -1/4
The reciprocal of x is 1/x
Just the same way, the reciprocal of 1/x is: x.
22
REAL NUMBERS, THE NUMBER LINE, ORDER RELATIONS, AND ABSOLUTE
VALUE – EXERCISES
1
7
1. For the set { −1, 0, , 4. 3, 7 , π , − π , 1, 7 , −18, − 2 , , −1. 8, 3 } list all of the
2
8
(a)
(b)
(c)
(d)
2.
counting numbers
integers
negative integers
rational numbers
real numbers
irrational numbers
even numbers
prime numbers
Locate the following numbers on a number line:
(a) 4
(d) −
3.
(e)
(f)
(g)
(h)
(b)
1
2
3
4
(e) –2.7
(c) –3
(f) 4.1
Fill in the appropriate sign (< or >):
(a) 5 ___ 1
(b) –1 ___ 3
(c) 0 ___ –2
(d) –3 ___ –7
(e) –2 ___ 6
(f) 5 ___ –4
4. Express the following without absolute value symbols:
(a) 8
(b) −1
(d) − 3
(e) − −5
−8
9
(f) − 2
(c)
5.
5
What is the reciprocal of − ?
4
6.
Which real number has no reciprocal?
7.
Which two real numbers are their own reciprocals?
8.
What is the reciprocal of
23
2?
REAL NUMBERS, THE NUMBER LINE, ORDER RELATIONS, AND ABSOLUTE VALUE –
ANSWERS TO EXERCISES
1.
(a) 1, 7, 3
(e) All numbers in the set are real numbers.
(b) –1, 0, 1, 7, –18, 3
(f)
(c) –1, –18
(g) 0, –18
7
1
(d) − 1, 0, , 4.3, 1, 7, − 18, , − 1.8, 3
8
2
(h) 7, 3
2.
CE
–4
(a) 4
3.
D
–3
–2
(b)
3
= 0.75
4
–1
7, π , − π , − 2
B
0
(c) –3
AF
1
2
(d) −
3
1
=
0.5
2
(a) 5 > 1
(b) –1 < 3
(d) –3 > –7
(e) –2 < 6
(c) 0 > –2
(f) 5 > –4
(b) −1 = 1
(d) − 3 = −3
4. (a) 8 = 8
(e) − −5 = −5
(c) −
8 8
=
9 9
4
(e) –2.7
(f)4.1
(f) − 2 = 2
4
5
5.
−
6.
0
7.
1, –1
8.
1
2
24
SCIENTIFIC NOTATION
Scientific notation refers to expressing a number as a product of any number between 1 and 10 to the
10th power. Scientific notation is mostly used when dealing with large quantities or numbers containing
many digits since it shortens the notation.
Examples:
Original number
→
Scientific Notation
76300
→
7.63 × 10 4
2,560,000
→
2.56 × 10 6
0.000066
→
6.6 × 10 −5
0.005
→
5 × 10 −3
To write a number in scientific notation:
If the number is in decimal notation, move the decimal point to the right of its original position and
place it after the first non-zero digit. The exponent of 10 will be the number of places the original
decimal point was moved, and it will be negative since it was moved to the right.
Examples:
0.0000643
→
6.43 × 10 −5
If the number to be changed to scientific notation is a whole number greater than 10, move the decimal
point to the left of its original position and place it after the first digit. The exponent of 10 will be the
number of places the original decimal point was moved, and it will be positive since it was moved to
the left.
Examples:
25
125,000
→
1.25 × 10 5
SCIENTIFIC NOTATION – EXERCISES
Express the following numbers in scientific notation:
1.
0.00125
2.
2,000,000,000
3.
796,000
4.
872
5.
90
6.
27 × 103
7.
281 × 102
8.
0.00179
9.
0.0000763
10. 367 × 10−3
26
SCIENTIFIC NOTATION -- ANSWERS TO EXERCISES
1.
0. 00125 = 1. 25 × 10−3
2.
2 , 000, 000, 000 = 2 × 109
3.
796, 000 = 7. 96 × 105
4.
872 = 8. 72 × 102
5.
90 = 9 × 101
6.
27 × 103 = 2. 7 × 104
7.
281 × 102 = 2. 81 × 104
8.
0. 00179 = 1. 79 × 10−3
9.
0. 0000763 = 7. 63 × 10−5
10.
367 × 10−3 = 3. 67 × 10−1
27
EXPONENTS
Exponents are used to write long multiplications in a short way. The exponent will tell you how many
times the number or letter needs to be multiplied.
(6)(6) = 6 2
Examples:
a ⋅ a ⋅ a ⋅ a = a4
In exponent notation, the number or letter being multiplied several times is called the base, and the
exponent, or the number that tells you how many times you need to multiply is called the power.
→
23
2 is the base, and 3 is the power
Exponents are mostly used when dealing with variables, or letters, since it is easier and simpler to
write x 4 than x ⋅ x ⋅ x ⋅ x .
There are a few rules used for simplifying exponents:
Zero Exponent Rule – Any number or letter raised to the zero power is always equal to 1.
Example:
30 = 1
a0 = 1
Product Rule – When multiplying the same base, the exponents add together.
Example:
← Same base, x, add the exponents, 3 + 4 = 7
x3 ⋅ x4
x3 ⋅ x4 = x7
Quotient Rule – When dividing the same base, subtract the exponents.
Example:
x5
x2
← Same base, x, subtract the exponents, 5 – 2 = 3
x5
= x3
2
x
28
Power Rule – When the operation contains parentheses, multiply the exponent on the inside with the
exponent on the outside.
Example:
(y )
6 2
(y )
6 2
← Multiply the exponents, 6 × 2 = 12
= y 12
When there is a fraction inside the parentheses, the exponent multiplies on the current power of the
numerator and the denominator. However, this rule does not apply if you have a sum or difference
within the parentheses, in that case a different rule will apply.
( )
( )
=
a2 + b2
a+b
Be careful: 
 is not the same as 2
c +d2
c+d 
!
 x3
 2
y
2
Examples:
32
9
3
  = 2 =
16
4
4
4

x3
 =
y2

2
!
4
4
x 12
y8
2
2
 a + b   a + 2ab + b 
=

  2
2 
 c + d   c + 2cd + d 
2
In fact:
Negative Signs – If the negative sign is outside the parentheses, perform the operations inside the
parentheses and carry out the negative sign to the final answer.
3
Example:
− ( 3) =
−(3)(3)(3) =
−(27) =
−27
However, if the negative sign is inside the parentheses, the negative sign will be affected by the
exponent.
Example:
(−3)3 =
(−3)(−3)(−3) =
−27
If the negative sign is inside the parentheses and the exponent is an even number, the answer will be
positive. If the exponent is an odd number, then the answer will be negative.
Examples:
− (2 ) = −8
3
← Since the negative sign is outside the parentheses, carry it
out to the final answer.
(− 5)2 = (− 5)(− 5) = 25
← Since the negative sign is inside the parentheses, it
needs to be carried out through the operation.
(− 4)2 = (− 4)(− 4) = 16
← Even number of exponents, positive answer.
(− 4)3 = (− 4)(− 4)(− 4) = −64
29
← Odd number of exponents, negative answer.
Negative Exponents – Whenever the problem, or the answer to the problem, contains negative
exponents, these exponents need to be made positive. An answer with negative exponents will most
likely be counted wrong. To change negative exponents into positive exponents, get the reciprocal
fraction. In simpler words, if the negative exponent is on the top, move it down; if the negative
exponent is on the bottom, move it up.
Examples:
x −2
← Get the reciprocal, or move the negative exponent down.
x −2
1
x −2 =
= 2
1
x
5 y −3
← Get the reciprocal of only the base with the negative exponent, the
number stays in its place.
5 y −3
5
= 3
5 y −3 =
1
y
a2
b −4
← Get the reciprocal of the base with the negative exponent, the base
with the positive exponent stays in its place.
a2
a2 ⋅ b4
=
= a2 ⋅ b4
1
b −4
30
RADICALS
For every operation there is an opposite or inverse that can be used to cancel the operation.
inverse
–
inverse
Example:
x
x2
+
3
3
inverse
inverse
x
x
×
÷
The opposite/inverse of exponents are called radicals. If the exponent is a square, or 2, the opposite
would be a square root; if the exponent is 3, the opposite would be a cubic root, and so on.
Examples:
22 = 4
23 = 8
35 = 243
inverse
inverse
inverse
3
5
4=2
8=2
243 = 3
Radicals can also be written in exponent notation, however, in this case the exponent would be a
fraction.
4 = 41 2
Examples:
3
8 = 81 3
5
243 = 2431 5
The expression 3 8 is called a radical expression, where 3 is called the index,
is the radical sign,
and 8 is called the radicand. The index of a radical expression must always be a positive integer
greater than 1. When no index is written it is assumed to be 2, or a square root.
Negative Radicals – The only restriction that exists for negative signs and radicals is that there cannot
be a negative sign under an even root since there is no real solution to this problem. However, you can
have a negative in front of a radical or under odd roots and still be able to obtain a real number.
Examples:
3
− 16
← Negative sign under even root, no real answer.
(4)(4) = 16 or
(− 4)(− 4) = 16
−8
← Negative sign under odd root, real answer.
(− 2)(− 2)(− 2) = −8
− 81
31
← Negative sign is outside the radical, the sign does not affect the
calculations, it is only carried out to the final answer.
− 81 = −(9 )(9 ) = −81
Radicals Containing Large Numbers – When working with large numbers under a radical sign, it is
always easier to break down the number and work on smaller part rather than trying to find an exact
root.
Example:
← Break down the number into smaller numbers.
525
21 ⋅
← Find the root of each of the smaller numbers.
25
← Write final answer with single digits first and radicals
at the end.
21 ⋅ 5
5 21
Sometimes it is not easy to find two small numbers that when multiplied give you the large numbers.
In this case, you may break down the large number little by little.
Example:
← Follow the same procedure as described above. It
takes a little longer, but at the end the same results are
obtained.
525
105 ⋅ 5
105 ⋅
5
21 ⋅ 5 ⋅
21 ⋅
5
25
21 ⋅ 5
5 21
32
LAWS OF EXPONENTS AND RADICALS
1. x m ⋅ x n = x m+ n
1
xn
3. x − n =
2. x 0 = 1 where x ≠ 0
4.
1
= xn
x −n
5.
xm
= x m− n
n
x
6.
xm
=1
xm
7.
(x )
8.
(xy )n = x n y n
m n
= x mn
n
x
10.  
 y
 x
xn
9.   = n
y
 y
1
n
11. x = x
12. x
x ⋅ n y = n xy
13.
n
15.
m n
17.
n
x =
( x)
m
m
mn
x
n
14.
n
−
1
n
−n
=
 y
= 
x
n
n
1
x
x
x
=n
y
y
m
n
16. x = n x m =
( x)
n
m
=x
Note 1: The division of any number by zero is undefined. Also, if x is negative and n is even,
x
1
n
are not defined.
Note 2:
x
is undefined
0
0 x = 0 for x ≠ 0
1⋅ x = x
33
1x = 1
0⋅ x = 0
x
=x
1
n
x and
EXPONENTS AND RADICALS - EXERCISES
Simplify:
1. x 5 ⋅ x 7
2. x 0 (for x ≠ 0 )
4. 5−3
5.
5
7.  
2
−2
5
8.  
2
x2 y
10.
xy
( )
4
13. − a 2
16.
(a b c ) (− ab)
2 2 2 3
 x3 
19.  2 
x 
3
4
4 −1 3
 8 
25.  
 27 
28.
31.
4
−
2
3
6.
(3x )2
9.
(x )
2
(for a + b ≠ 0)
3 2
(3 )
12. 9 ⋅ 9 2
14.
(− a )
15.
c7 ⋅ c ⋅ c2
c5 ⋅ c 2 ⋅ c 3
17.
(a )
18.
(4a b) (3a b )
21.
(a x )
2 −2
2 4
−1 −1
1
20.  
k
(a x )
(a b )
(a + b )0
11.
2 −3 2
22.
22
23
3.
−2
2
2
(− 1)5
24.
( −1)
26.
27
27.
72
3
2
 1 2
30.  
4
29. 9
50
32.
3
3 3
2 −3 2
23.
16
2
9
7
250
34
EXPONENTS AND RADICALS - ANSWERS TO EXERCISES
1. x 5 ⋅ x 7 = x 12
2. x 0 = 1
3.
(a + b )0 = 1
4. 5−3 =
5.
22
1
−3
−1
22=
2=
=
3
2
2
6.
−2
x )
(=
11.
(3 )
3
2 −2
= 3− 4 =
( )
13. − a 2
10.
1
1
=
4
3
81
=
− a2 a2 a2 a2 =
−a8
c 7 ⋅ c ⋅ c 2 c10
15. 5 2 3 = 10 = 1
c ⋅c ⋅c
c
17.
(a )
−1 −1
=a
21.
(a x )
23.
( −1)
5
4 −6
=a x
27.
16.
(a b c ) (− ab)
= a10b10c 6
18.
(4a b) (3a b )
= 432a10b11
2
4
2
2 2 2 3
2
4
2
2
−2
22.
(a x )
(a b )
24.
( −1)
2 −3 2
a4
= 6
x
 8 
3
=

 27 
 8 
25. =
 
 27 
a
( −a ) =
( −a )( −a )( −a )( −a ) =
2 3 3
4 −1 3
9
=
31.
35
8
a 4 x−6
b3
=
a12b − 3 a8 x 6
=
−1
27 = 3 ⋅ 9 = 3 3
26.
28.
4
16 = 2
7
29. 9
2
−2
−2
 38 
9
2
=
=
 3

 
4
3
 27 
72 = 36 ⋅ 2 = 6 2
32
2
1−2
1
20.  = =
k2
−2
k
k
=
−1
2
−
3
14.
−2
3
 x3 
x9
19.  2 =
=
x3

6
x
x 
2 −3 2
x2 y
= x( 2−1) ⋅ y (1−1) = x1 ⋅ y 0 = x
xy
12. 9 ⋅ 9 2 = 9 3
( )( )( )( )
4
= 32 ⋅ x 2 = 9 x 2
52 25
5
8.  = =
22
4
2
( 3)( 2 )
x=
x6
2
2
2
2
22
4
5
2
7.  =  = =
2
5
25
2
5
9.
( 3x )
1
1
=
3
5 125
= 9 = 81 ⋅ 9 = 81 ⋅ 9 = 9 ⋅ 3 = 27
3
50 = 25 ⋅ 2 = 5 2
7
7
 1   1 7
1
 1 2
1
 =  =
30.   =   = 

4
4
 4   2  128
32.
3
250 = 3 125 ⋅ 2 = 53 2