Midterm power point

For your mid-term, be certain you understand the following: 1) Real numbers 2) Rational numbers 3) Irrational numbers 4) Whole numbers 5) Integers 6) Negative numbers

You will be asked a LOT of questions where you’ll need to understand the nuances between these sets of numbers.

1) Prime, Composite or neither? 9

Prime numbers are numbers that nothing can be divided into, you can’t break them down at all (except for dividing them by 1 and themselves).

So if you can think of ANY number that goes into 9, you’d have to say that number is composite.

I know 3 will go evenly into 9. So 9 must be a composite number.

composite

2) Write the prime factorization of 98

Prime factorization: To work Quotient of 1, start with the smallest prime, and work your way up. 2, 3, 5, 7, 11

¬ ¬

¬

98 2

49 7

7 7

1

2 ⋅ 7 ⋅ 7    𝑜𝑟   2 ⋅ 72

3) Reduce. 24

40

You have to think of numbers that go into both the numerator and the denominator, and then divide both by that number!

4 will go into both numbers, so divide them by 4:

6

10

24

40

Can you still think of a number that goes evenly into both? Yes! Divide again by 2!

6 3

10 5

You could do this problem in only one step, IF you’d seen that 8 would divide both numbers. Either way, we get to the same answer!

3

5

4) Reduce. 36

42

You have to think of numbers that go into both the numerator and the denominator, and then divide both by that number!

2 will go into both numbers, so divide them by 2:

18

21

36

42

Can you still think of a number that goes evenly into both? Yes! Divide again by 3!

18 6

21 7You could do this problem

in only one step, IF you’d seen that 8 would divide both numbers. Either way, we get to the same answer!

6

7

5) Divide. 3

3 57

If you’re gonna divide fractions, first make them both fractions.

This is division, so multiply the reciprocal.

24 5

7 1

24 1

7 5

Cancel what you can, nothing in this case. So multiply across.

24 1 24

7 5 35

24

35

6) Multiply. 2 3

9 75 5

If you’re gonna multiply fractions, first make them both fractions.

47 38

5 5

Cancel what you can. Nothing here to cancel. Then multiply.

47 38

5 5

1786

25

And if you need to, change to a mixed number.

1786 1171

25 25or

1171

25

7) Divide. 14 1

415 2

If you’re gonna divide fractions, first make them both fractions.

This is division, so multiply the reciprocal.

14 9

15 2

14 2

15 9

Cancel what you can, nothing in this case. So multiply across.

28

135

14 2 28

15 9 135

8) Add and simplify. 13 3

26 26

ALWAYS add or subtract vertically:

13

26

3

26

16 8

26 13

And reduce

8

13

9) Find the sum or difference. Write the answer in lowest terms.

2 1

5 15

ALWAYS add or subtract vertically:

     2

5=

6

15

+1

15=

1

15

7

15

Make sure to reduce your answer, if possible.

7

15

10) Find the value of the exponential expression.

222

This means 22 x 22

484

11) Simplify. 7 7 6 6 This is an Order of Operations problem: PEMDAS rules!

1) Parenthesis? No, move on. 2) Exponents?

3) Multiplication or division, in order from left to right. Yep, do that. 49 - 36 4) Addition or subtraction, in order from left to right. 13

13

12) Simplify. 24(3 ) 8(4 6) ( 9)

This is an Order of Operations problem: PEMDAS rules!

1) Parenthesis? Yes: 4(9) + 8 (10) –(-9) 2) Exponents? Not anymore. =) 3) Multiplication or division, in order from left to right. Yep, do that. 36 +80 +9 (double negative turns into a positive) 4) Addition or subtraction, in order from left to right. 125

125

13) Simplify. 22 ( 5 3) 12 ( 1)

7 6 ( 2 4) (6 7)

Step by step, take care of the numerator and denominator separately:

22 ( 5 3) 12 ( 1)

22 ( 8) 12 ( 1)

PEMDAS!!!

4 ( 8) 12 ( 1)

32 12

44

7 6 ( 2 4) (6 7)

7 6 ( 8) (42)

7 48 (42)

41 (42)

1

Bring them back together: 44

441

-44

14) True or false?

2 2 210 5 2

100 25 4

100 21

2 2 210 5 2

False

15) Simplify. 4.4 6.16 2.8 4.5

This is an Order of Operations problem: PEMDAS rules!

1) Parenthesis? No. 2) Exponents? No. 3) Multiplication or division, in order from left to right. Yep, do that. 4.4-2.2+4.5 4) Addition or subtraction, in order from left to right. 6.7

6.7

16) Calculate. 35 3 3 7 This is an Order of Operations problem: PEMDAS rules!

1) Parenthesis? No. 2) Exponents? No. 3) Multiplication or division, in order from left to right. Yep, do that. 35-9+7 4) Addition or subtraction, in order from left to right. 33

33

17) Find the value of the expression. 33 36 9 5 This is an Order of Operations problem: PEMDAS rules!

1) Parenthesis? No. 2) Exponents? No. 3) Multiplication or division, IN ORDER from left to right. Yep, do that. (tricky!!!) 4) Addition or subtraction, in order from left to right. 53

33 4 5

33 20

53

18) Translate: Twelve is greater than three plus eight.

Twelve 12 Twelve is greater than 12> Twelve is greater than three 12>3 Twelve is greater than three plus 12>3+ Twelve is greater than three plus eight 12>3+8

12>3+8

19) Evaluate for b = 3. 29b

Plug in, then remember Order of Operations! Exponents first! Then multiplication.

2

9 3

9(9)

81

81

20) Translate: The quotient of a number and two.

The quotient The quotient of a number The quotient of a number and two

x

2

x

21) Decide if the given number is a solution to the given equation.

6 3 23; 3m

Plug in! 6(3) + 3 =23 18 +3 =23 21 = 23 NO!

NO

22) Identify as an expression or an equation.

7c d

Does it have an equal sign? No, so it’s NOT an equation. Must be an expression.

expression

Naturals: 1, 2, 3 . . . Wholes: 0, 1, 2, 3 . . . Integers: . . . -2, -1, 0, 1, 2, 3 . . . Rationals: anything written as a fraction Irrationals: Reals: for you, all the numbers there are

2, 3, 5 ...

23) List all the numbers from the given set that are of each category:

35, ,0,0.5, 2,7.7, 16

7

Be careful: is really 4

16

a) Naturals: , cuz he’s really 4 b) Wholes: 0, c) Integers: -5, 0, d) Rationals: all of them except e) Irrationals: f) Reals: all of them

1616

16

22

24) True or false: Some rational numbers are irrational.

False. Rational and irrational are opposites. A number is EITHER a rational or an irrational . . . can’t be both.

False

Naturals: 1, 2, 3 . . . Wholes: 0, 1, 2, 3 . . . Integers: . . . -2, -1, 0, 1, 2, 3 . . . Rationals: anything written as a fraction Irrationals: Reals: for you, all the numbers there are

2, 3, 5 ...

25) True or false: Every whole number is a real number.

True. At this point, ALL numbers you see are real numbers, so it doesn’t matter what you’re looking at : it’s real!

True

Naturals: 1, 2, 3 . . . Wholes: 0, 1, 2, 3 . . . Integers: . . . -2, -1, 0, 1, 2, 3 . . . Rationals: anything written as a fraction Irrationals: Reals: for you, all the numbers there are

2, 3, 5 ...

26) True or false: Every integer is an irrational number.

Naturals: 1, 2, 3 . . . Wholes: 0, 1, 2, 3 . . . Integers: . . . -2, -1, 0, 1, 2, 3 . . . Rationals: anything written as a fraction Irrationals: Reals: for you, all the numbers there are

2, 3, 5 ...

False. In fact, every integer is a rational number because all integers can be written as a ratio . . . just put a 1 in the denominator!

A true statement here would have been: “Every integer is a rational number.”

False

27) True or false: The absolute value of any nonzero number is positive.

True. ALL absolute values end up being positive (or zero) because absolute value asks you “How far away is that number from zero?”

5

3

243

3

5

All of these end up being positive numbers

True

28) Select the smaller of the two given numbers.

5 1,

8 6

Compare by first getting a common denominator. 5 15

8 24

1 4

6 24

So you’re comparing to 15

24

4

24

Look only at the numerators. Which is smaller, -15 or -4?

-15 is, so the first fraction is the smaller

5

8

29) Find the absolute value. 15

How far away from 0 is -15 on the number line?

-15

Absolute value means how far away is that number from 0. In short cut, it just means take any negative sign off the number.

And don’t forget to drag that negative sign back in there!

That first negative sign isn’t part of the action. He’s just waiting for you to figure out what you’re gonna do with the absolute value part.

30) Add the following. 78 ( 70)

The signs are the same so we’ll add the two numbers and take their sign.

This is like I owe you 78 and I owe you $70. I end up owing you $148

148

31) Add the following. 29 ( 11)

The signs are different so you subtract the easiest way you can

29 11 18

And take the sign of the larger number, so 18 is positive.

18

32) Subtract the following: -2 -7

The signs are the same so we’ll add the two numbers and take their sign. Remember: the sign in front of the number belongs to the number!

This is like I owe you $2 and I owe you $7. I end up owing you $9

-9

33) Find the difference.

Minus a minus? Always change to plus a plus!

28

13 ( 15)

13 ( 15)

13 ( 15)

28

34) Use order of operations to simplify the expression.

1) Parenthesis? Yes!

2) Exponents? Nope, none of those. 3) Multiplication or division, in order from left to right. Nope 4) Addition or subtraction, in order from left to right. 8

This is an Order of Operations problem: PEMDAS rules!

8

20 [(11 20) ( 19 18)]

20 [( 9) ( 37)]

20 [( 9) ( 37)]

20 [28]

35) Translate: The quotient of a number and 3 is -2.

The quotient The quotient of a number The quotient of a number and 3 The quotient of a number and 3 is -2.

x

3

x

23

x

And x = -6