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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 𝑥2 = 𝑝 and 𝑥3 = 𝑝?

Name ________________________________________________

Period __________

Team ________________

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Number Sets

All numbers can be classified into number sets.

Integers

Rational Numbers

Irrational Numbers

Real Numbers

Examples:

Examples:

Examples:

Examples:

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THE REAL NUMBER SYSTEM

Place a check mark () in all number category columns to which each number belongs.

Number Whole Number Integer Rational Number

Irrational Number

Real Numbers

5

-13

2.79

½

0

√𝟐𝟓

√𝟐𝟑

𝟎. �̅�

π/2

State if the decimal terminates, repeats or neither. Then identify each number as rational or irrational.

1) 6 2) 3

7 3) 𝜋 4) 9.381

5) −250 6) √3 7) 0.141414 … 8) −1

3

9) √49 10) 52.173916 … 11) 0 12) −5.72

Real Numbers

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Use your calculator to write each rational number as a decimal (rounding decimals to the nearest thousandths.)

13) 5

8 14)

3

5 15)

2

3 16) 3

12

25

17) −29

60 18) 83

721

999 19)

4

625 20) −

2

11

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. 43̅̅̅̅

25) 9.98 26) 1. 513̅̅ ̅̅ ̅ 27) 0.87 28) −32. 05̅̅̅̅ 29) Which number is an integer?

a. −115

b. −7 c. √15 d. 1

2

30) Which number is a whole number?

a. 5

6 b. −4 c. √36 d.

1

4

31) Which number is irrational?

a. 9. 27̅̅̅̅ b. √2 c. 5√9 d. −37

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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 ________

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

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

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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.

√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) 13 feet B) 12 feet C) 11 feet D) 10 feet

11) What are two values for a & b that would satisfy this diagram?

A) a = 9.2, b = 9.8 B) a = 81, b = 100 C) a = 85, b = 96 D) a = 95, b = 99

D E F G H I J K L M

A

N O P Q R A B C

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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) −71

2______ − 8 11) 12.999_______13 12) √19_______4. 8̅

13) 7.2_______√52 14) −√8_________ − 2. 63̅̅̅̅ 15) 11

3_______√10

Write each set of numbers in order from least to greatest.

16) 12

5, √6, 2. 4̅,

61

25 17) 2. 71̅̅̅̅ , −√7 , 2

2

3,

53

−20

__________________________________ __________________________________ Graph each set of numbers.

18) {−7, 31

2, √25, − 2.08, √10}

19) {−41

3, − √16,

2

5, − 3.5, 0. 85̅̅̅̅ }

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Cube Roots

A perfect cube is a number that has an integer as its cube root. For example,

√83

= ________

√−273

= ________ So, 8 and -27 are perfect cubes. List the perfect cubes from 1 to 125 on the line below: __________________________________________________________________________________ Find each cube root.

1) √643

= ___________ 2) √13

= ___________

3) √−13

= ___________ 4) √−1253

= ___________ Not all numbers are perfect cubes. To find these cube roots, use your calculator to estimate the value.

5) √683

= ___________ 6) √263

= ___________

7) √−93

= __________ 8) √43

= ___________ Replace each _______ with a >, < or = to make each sentence true. You may use a calculator.

9) √1503

_______6 10) 22

7 _______√64

3 11) −3 _______√−30

3

10) √8______√83

11) √1003

_______4. 6̅ 12) −√49_______√−3433

Write each set of numbers in order from least to greatest.

13) 31

3, √27

3, √7,

31

15

_____________________________________________

14) √64, √643

, −√64, √−643

____________________________________________

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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 √ and √ 3

, you can also solve equations with the x raised to the power of 2 or 3. 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) 𝑥2 = 100 3) 𝑘2 = 9 Check #1: 4) 𝑐2 = 24 5) 𝑣2 = 90 6) 𝑐2 = 7 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.

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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) 𝑦3 = −1 9) 𝑎3 = 125 Check #7: 10) 𝑝3 = 50 11) 𝑧3 = −25 12) 𝑢3 = −210 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) 𝑘2 = 35 14) 𝑥3 = −8 15) 𝑐2 = 49 16) 𝑥3 = 12

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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]

4) 4𝑎

𝑎+𝑏 5) 𝑐(20 + 2𝑎)

Solve the following equations. You must show work in steps. Circle the answer. [U2, pg 2-5, 9-12]

6) −80 = −3𝑥 + 43 7) 2

9𝑐 − 5 = 13

8) 5(3𝑤 + 7) = 65 9) 5𝑦 + 2(𝑦 + 8) = −40

10) 7𝑚 + 13 = 4𝑚 + 8 11) −5𝑚 − 7 = −5𝑚 + 13

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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 1

√𝑥> √𝑥 > 𝑥 true?

A) 1

2 B) 0 C) -2 D) 3

3) For what value of 𝑛 is √𝑛 < 1 <1

√𝑛 true?

A) -5 B) 1

5 C) 0 D) 5

4) The time it takes for a falling object to travel a certain distance 𝑑 is given by the equation, 𝑡 = √𝑑

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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

and found them to be 433

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) |3

7|

10) |2.6 + 1.8| 11) −|9| 12) −|−2| 13) − |5

12−

1

4|