Unit 2Integers,

Irrationals, Squares, Cubes

and RootsPre-Algebra 8

HEADINGName ___________________Date _________ Period _____

Perfect Squares

MEMORIZE!

12 = 112 =

22 = 122 =

32 = 132 = 252 =

42 = 142 = 302 =

52 = 152 = 402 =

62 = 162 = 502 =

72 = 172 = 602 =

82 = 182 = 702 =

92 = 192 = 802 =

102 = 202 = 902 =

Perfect Cubes

MEMORIZE!

13 = 63 =

23 = 73 =

33 = 83 =

43 = 93 =

53 = 103 =

Adding Rationals (mainly integers)

Hints!

RAP!

Use a number line to model the sum of two integers. 1. 2. Find the sum. 3. 5. 7.

4. 6. 8.

Same signs …ADD! Signs opposite … SUBTRACT! (Change doubles!)

Use a number line, $ money $ , or the RAP.-2 -1 0 1 2 3

Evaluate.

9. 10. 11.

12. A scuba diver studying marine life is 4 feet below sea level. From that depth, the diver descends 72 feet to the ocean floor and then rises 61 feet. The diver rests there to avoid decompression illness. Where is the diver relative to sea level?

13. Jason got the following scores on his first 9 holes of miniature golfing. -2, 3, 0, 1, -1, -1, 2, 4, 1 What was his score after the 9th hole?

Subtracting Rationals (mainly integers)

Hints!

RAP!

Simplify. Evaluate.

1. 7. 9.

2.

3. 8. 10.

4.

5.

Same signs …ADD! Signs opposite … SUBTRACT! (Change doubles!)

Use a number line, $ money $ , or the RAP.-2 -1 0 1 2 3

11. The temperature in Havre, Montana yesterday was –12 degrees. During a Chinook the temperature will drop 20 more degrees, but then warm up 50 degrees. What will the temperature be after the Chinook?

12. Your checking account shows an overdraft, or a negative balance. Your present balance is -$25. You deposit $100, then write a check for $12. What is your new balance?

13. On its first down, the football team gains 8 yards. On its second down, it gains 1 yard. On its third down, it loses 5 yards. On its fourth down, it gains 7 yards. Does the football team gain at least 10 yards?

Multiplying and Dividing Rationals

HAND SIGNAL

Simplify.

1. 7. 10.

2.

3. 8.

4. 11.

5. 9.

6.

One negative makes a negative

Two negatives make a positive

Division with ZERO THINK ABOUT THE EGG SCENARIO!!Zero Divided by a Number = A Number Divided by Zero =

Proof: Proof:

Answer the following. It will be helpful to write the division problems as fractions.

12. 13. 14.

BE CAREFUL when it comes to negatives and exponents!The expression -32 means the opposite of 32. The exponent 2 applies to the base 3. The negative sign is not part of the base. In the expression (-3)2, the negative sign is part of the base -3. The exponent 2 applies to the base -3.

Use the order of operations to simplify each expression.

15. 17 19.

16. 18. 20.

Square Roots Squaring and Square Rooting are INVERSE OPERATIONS. They undo one another.

THINK! If 102 = 100 then = 10

Square Root of a Number:

Radical Symbol: Radicand:

Special Cases:

16

16

16

0

16

If b2 = a, then b is a square root of a

The number inside the radical symbol

The (positive) square root of 16.It is known as the principal square root. The negative square root of 16.

This is read “plus or minus the square root of 16”.

Zero has only one square root.

Negative numbers have NO REAL SQUARE ROOTS. A number times itself will NEVER be negative!

Examples:

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

100

A radical expression is in simplest form when:No factor of the expression inside the radical sign has any perfect square factor other than 1.

There are no fractions under the radical sign, and no radical sign in the denominator of any fraction.

Quotient Property

Use the Quotient Property to simplify.

11. 13. 15.

12. 14. 16.

Approximate the value of a non-perfect square root:

Steps: 1)

2)

3)

1. 4.

2. 5.

3. 6.

Find the two closest perfect squares (greater than and less than)See where your number fits in

Make a decimal approximation (nearest tenth)

Plot Irrational Numbers on a Number Line

7. 8.

Order and Order Rational and Irrational Numbers

9. Compare 10. Compare using <, >, or =.and make comparative statements.

11. Order the numbers from least to greatest: HINTS! *change all to decimals *use a number line

50 and 60

Simplifying Square Roots:

An expression is in simplest form when:

No factor of the expression inside the radical sign has any perfect square factor other than 1.

There are no fractions under the radical sign, and no radical sign in the denominator of any fraction.

Product Property Quotient Property

* EASY METHOD FOR SIMPLIFYING SQUARE ROOTS:

Examples:

1. 3. 5.

2. 4. 6.

Siberian Prison Story:

_______ escape, but only _______ survivesFactor the radicand completely and pull out pairs. Leave “singles” in the radical.

Simplify Square Roots

Use the Order of Operations to simplify. * Radicals are grouping symbols

7. 10. 13.

8. 11. 14.

9. 12. 15.

Perfect Cubes and Cube Roots

For PERFECT CUBES 1 – 10, see back of flipbook

THINK ABOUT NEGATIVES!

(-1)3 = (-3)3 = (-5)3 = (-7)3 = (-9)3 =

(-2)3 = (-4)3 = (-6)3 = (-8)3 = (-10)3 =

Cube Roots Cubing and Cube Rooting are INVERSE OPERATIONS. They undo one another.

THINK! If 23 = 8 then = 2

You can take cube roots of negative numbers because a ______ times a _____ times a _____ equals a _______.

Examples:

1. 3. 5.

2. 4. . 6.

3

3 8

7. For which equation is

a solution?

A. C.

B. D.

3 22x 3 22x

3 66x

3 22

3x

3 322x