Unit 3 Rational and Irrational Numbers
How do I determine if a number is rational or irrational?
How do I write a fraction as a decimal?
How do I convert a terminating or repeating decimal into a rational number?
How do I estimate the value of an irrational number?
How do I compare, order, and graph rational and irrational numbers?
How do I solve equations in the form 𝑥 2 = 𝑝 and 𝑥 3 = 𝑝?
Name ________________________________________________
Period __________
Team ________________
1
Number Sets
All numbers can be classified into number sets.
Integers
Examples:
Rational Numbers
Examples:
Irrational Numbers
Examples:
Real Numbers
Examples:
2
THE REAL NUMBER SYSTEM
Real Numbers
Place a check mark () in all number category columns to which each number belongs.
Rational
Irrational
Number Whole Number
Integer
Real Numbers
Number
Number
5
-13
2.79
½
0
√𝟐𝟓
√𝟐𝟑
̅
𝟎. 𝟑
π/2
State if the decimal terminates, repeats or neither. Then identify each number as rational or irrational.
3
1) 6
2) 7
3) 𝜋
4) 9.381
1
5) −250
6) √3
7) 0.141414 …
8) − 3
9) √49
10) 52.173916 …
11) 0
12) −5.72
3
Use your calculator to write each rational number as a decimal (rounding decimals to the nearest
thousandths.)
5
3
2
12
13) 8
14) 5
15) 3
16) 3 25
29
17) − 60
721
18) 83 999
19)
4
2
20) − 11
625
Use your calculator to write each decimal as a fraction. If it is a repeating decimal, enter 9 or more
decimal places.
̅̅̅
21) 0.4
22) 0.005
23) 0. 3̅
24) 5. ̅43
25) 9.98
29)
30)
31)
26) 1. ̅̅̅̅̅
513
Which number is an integer?
11
a. − 5
b. −7
Which number is a whole number?
5
a.
b. −4
6
̅̅
28) −32. ̅̅
05
27) 0.87
c. √15
d.
c. √36
d.
c. 5√9
d.
Which number is irrational?
̅̅̅
a. 9. ̅27
b. √2
4
1
2
1
4
−37
41
Estimating Square Roots
A perfect square is a number that has an integer as its square root. For example,
√16 = ________
√25 = ________
So, 16 and 25 are perfect squares. List the perfect squares from 1 to 144 on the line below:
__________________________________________________________________________________
You can use perfect squares to estimate square roots that are irrational.
√19 = between _______ and _______, but closer to ________
√23 = between _______ and _______, but closer to ________
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Without a calculator, estimate the values of the following square roots. State the 2 consecutive
integer values the answer lies between and then circle the integer closest to the answer. Write your
answers as modeled in example 1.
1) √82
2) √45
between _______ and ________
between _______ and ________
_______
________
_______
3) √140
4) √96
5) √6
6) √38
5
________
Answer these questions without a calculator.
7) Which of the following values is closest to the answer of √27?
A) 4.9
B) 5.2
C) 5.8
D) 13.5
8) Write the letter for the point which is closest to each square root.
A
B
C D
E F GH
I
J
K
L
MN O
P
Q
A
√17 _________
√35 _________
√25 _________
√3 _________
√9 _________
√58 _________
√64 _________
√83 _________
√20 _________
9) Which irrational number is between 5 and 6?
A) √12
B) √20
C) √34
D) √80
10) A square poster has an area of 152 feet2. Estimate the side length of the poster?
A)
B)
C)
D)
13 feet
12 feet
11 feet
10 feet
11) What are two values for a & b that would satisfy this diagram?
A)
B)
C)
D)
a = 9.2, b = 9.8
a = 81, b = 100
a = 85, b = 96
a = 95, b = 99
6
R
Compare, Order and Graph Real Numbers
Replace each _______ with a >, < or = to make each sentence true. You may use a calculator.
7) −4 _______ − 1
8) −5 _______ − 6
9) −9________2
10) −7 2 ______ − 8
11) 12.999_______13
12) √19_______4. 8̅
13) 7.2_______√52
̅̅̅
14) −√8_________ − 2. ̅63
15)
1
11
3
_______√10
Write each set of numbers in order from least to greatest.
16)
12
√6,
,
5
61
2. 4̅,
̅̅̅̅,
17) 2. 71
25
__________________________________
1
18) {−7, 3 2 , √25 , − 2.08, √10}
1
− √16,
2
5
, − 3.5,
2
2 3,
53
−20
__________________________________
Graph each set of numbers.
19) {−4 3 ,
−√7 ,
̅̅̅̅}
0. 85
7
Cube Roots
A perfect cube is a number that has an integer as its cube root. For example,
3
√8 = ________
3
√−27 = ________
So, 8 and -27 are perfect cubes. List the perfect cubes from 1 to 125 on the line below:
__________________________________________________________________________________
Find each cube root.
3
1) √64 = ___________
3
2) √1 = ___________
3
3
4) √−125 = ___________
3) √−1 = ___________
Not all numbers are perfect cubes. To find these cube roots, use your calculator to estimate the value.
3
3
5) √68 = ___________
6) √26 = ___________
3
3
7) √−9 = __________
8) √4 = ___________
Replace each _______ with a >, < or = to make each sentence true. You may use a calculator.
3
9) √150 _______6
10)
22
7
3
_______√64
11) √100_______4. 6̅
3
3
10) √8______√8
3
11) −3 _______√−30
3
12) −√49_______√−343
Write each set of numbers in order from least to greatest.
13)
1
3 3,
3
√27,
√7,
31
15
_____________________________________________
14) √64,
3
√64 ,
−√64,
3
√−64
____________________________________________
8
Solve 𝒙𝟐 = 𝒑 and 𝒙𝟑 = 𝒑
You have already learned how to solve equations such as:
2𝑥 + 8 = −6
and
4𝑥 − 6 = 𝑥 + 1.
In these equations (and all of the ones you have ever solved), the power of x is ________.
Now that you understand √
of 2 or 3.
3
and √
, you can also solve equations with the x raised to the power
Solve. For irrational answers, write the answer in both exact and estimated form (rounded to the
nearest thousandth (3 decimal places.))
1) 𝑛2 = 64
2) 𝑥 2 = 100
3) 𝑘 2 = 9
5) 𝑣 2 = 90
6) 𝑐 2 = 7
Check #1:
4) 𝑐 2 = 24
Check #4:
All equations with the variable squared have ____________ solutions.
The solutions are ________________________ of each other, so one is _______________________
and the other is ________________. We can use the symbol, _____________ to indicate both
solutions.
9
Solve. For irrational answers, write the answer in both exact and estimated form (rounded to the
nearest thousandth (3 decimal places.))
7) 𝑛3 = 64
8) 𝑦 3 = −1
9) 𝑎3 = 125
11) 𝑧 3 = −25
12) 𝑢3 = −210
Check #7:
10) 𝑝3 = 50
Check #10:
All equations with the variable cubed have ____________ solution.
The solution is the _______________________ sign as the number in the equation.
Your turn now…all mixed up. Solve.
13) 𝑘 2 = 35
14) 𝑥 3 = −8
15) 𝑐 2 = 49
16) 𝑥 3 = 12
10
Name__________________________________________
Period ________
UNIT 3 CUMULATIVE REVIEW
This page is mandatory. You must complete problems #1-11. You can work these problems anytime
throughout the unit, but it is due the day after we take the unit test.
Write an algebraic expression for each verbal expression. [U1, pg 3-4]
1) the product of 3 and b, decreased by 40
_________________
2) 8 less than twice w
_________________
3) the sum of 52 and k cubed
_________________
Evaluate each expression if 𝑎 = −5, 𝑏 = 3 𝑎𝑛𝑑 𝑐 = −2. You must show work in steps, even if you
used your calculator. Circle the answer. [U1, pg 11]
4𝑎
4)
5) 𝑐(20 + 2𝑎)
𝑎+𝑏
Solve the following equations. You must show work in steps. Circle the answer. [U2, pg 2-5, 9-12]
2
6) −80 = −3𝑥 + 43
7)
8) 5(3𝑤 + 7) = 65
9) 5𝑦 + 2(𝑦 + 8) = −40
10) 7𝑚 + 13 = 4𝑚 + 8
11) −5𝑚 − 7 = −5𝑚 + 13
11
9
𝑐 − 5 = 13
BONUS PAGE
This page is not mandatory but can be used for additional challenge work at any time during the unit.
1) Which sentence is true?
A) All real numbers are irrational numbers.
B) All integers are rational numbers.
C) All rational numbers are integers.
2) For what value of 𝑥 is
A)
1
2
1
√𝑥
> √𝑥 > 𝑥 true?
B) 0
3) For what value of 𝑛 is √𝑛 < 1 <
A) -5
B)
1
5
1
√𝑛
C) -2
D) 3
C) 0
D) 5
true?
𝑑
4) The time it takes for a falling object to travel a certain distance 𝑑 is given by the equation, 𝑡 = √16
where 𝑡 is in seconds and 𝑑 is in feet. If Krista dropped a ball from a window 28 feet above the
ground, how long would it take for the ball to reach the ground?
5) Police can use the formula 𝑠 = √24𝑑 , to estimate the speed 𝑠 of a car in miles per hour by
measuring the distance 𝑑 in feet a car skids on a dry road. On his way to work, Jerome skidded
trying to stop for a red light and was involved in a minor accident. He told the police officer that he
was driving within the speed limit of 35 miles per hour. The police officer measured the skid marks
3
and found them to be 43 4 feet long. Should the officer give Jerome a ticket for speeding? Explain.
Absolute Value of a number is its distance from zero on a number line. Absolute value of a number n is
written as |𝑛|. The absolute value bars act as grouping symbols, so do any math problem inside first.
Example A) Simplify |16|
Example B) Simplify |−7|
answer: 16
answer: 7
Simplify.
6) |−23|
7) |6|
8)
|15 − 3|
9)
10) |2.6 + 1.8|
11) −|9|
12)
−|−2|
13)
12
3
|7|
5
1
− |12 − 4|