place the letter of each value in its location in the real number system below.
a. -0.(overline{2}) b. 18
c. (-sqrt{100}) d. (pi)
e. 0 f. (2\frac{1}{6})
g. -5 h. 4.03
i. (-sqrt{72}) j. (sqrt{\frac{4}{9}})
k. (\frac{36}{9})
chart: irrational circle, rational circle with nested integers, whole, natural circles
topic 1: properties
identify the property shown below.
7. (4 + (x + y) = (4 + x) + y)
8. (\frac{2}{5} cdot \frac{5}{2} = 1)
9. if (sqrt{49} = 7) and (7 = 3 + 4), then (sqrt{49} = 3 + 4)
10. (-28 = -28)
11. (8x^2 cdot 1 = 8x^2)
12. (10y + (-10y) = 0)
13. ((a + 4) cdot 0 = 0)
14. (-5(x + 7) = -5x - 35)
15. ((x + 2) + y = (2 + x) + y)
16. if (x = -1), then (-1 = x)

Question

place the letter of each value in its location in the real number system below.
a. -0.(overline{2})  b. 18
c. (-sqrt{100})  d. (pi)
e. 0  f. (2\frac{1}{6})
g. -5  h. 4.03
i. (-sqrt{72})  j. (sqrt{\frac{4}{9}})
k. (\frac{36}{9})
chart: irrational circle, rational circle with nested integers, whole, natural circles
topic 1: properties
identify the property shown below.
7. (4 + (x + y) = (4 + x) + y)
8. (\frac{2}{5} cdot \frac{5}{2} = 1)
9. if (sqrt{49} = 7) and (7 = 3 + 4), then (sqrt{49} = 3 + 4)
10. (-28 = -28)
11. (8x^2 cdot 1 = 8x^2)
12. (10y + (-10y) = 0)
13. ((a + 4) cdot 0 = 0)
14. (-5(x + 7) = -5x - 35)
15. ((x + 2) + y = (2 + x) + y)
16. if (x = -1), then (-1 = x)

Accepted Answer

### Problem 7: Identify the property for $ 4 + (x + y) = (4 + x) + y $
# Explanation:
## Step1: Recall properties of addition
The associative property of addition states that for any real numbers $ a $, $ b $, and $ c $, $ a+(b + c)=(a + b)+c $.
## Step2: Match with the given equation
In the equation $ 4+(x + y)=(4 + x)+y $, we have $ a = 4 $, $ b=x $, and $ c = y $, which fits the associative property of addition.
# Answer: Associative Property of Addition

### Problem 8: Identify the property for $ \frac{2}{5}\cdot\frac{5}{2}=1 $
# Explanation:
## Step1: Recall properties of multiplication
The multiplicative inverse property states that for a non - zero real number $ a $, $ a\times\frac{1}{a}=1 $ (or $ \frac{a}{b}\times\frac{b}{a}=1 $ where $ a
eq0 $ and $ b
eq0 $). Here, $ \frac{2}{5} $ and $ \frac{5}{2} $ are multiplicative inverses of each other.
## Step2: Confirm the property
Since the product of $ \frac{2}{5} $ and $ \frac{5}{2} $ is 1, this is the multiplicative inverse property (also called the reciprocal property).
# Answer: Multiplicative Inverse Property (or Reciprocal Property)

### Problem 9: Identify the property for "If $ \sqrt{49}=7 $ and $ 7 = 3 + 4 $, then $ \sqrt{49}=3 + 4 $"
# Explanation:
## Step1: Recall properties of equality
The transitive property of equality states that if $ a=b $ and $ b = c $, then $ a=c $.
## Step2: Match with the given statement
Here, $ a=\sqrt{49} $, $ b = 7 $, and $ c=3 + 4 $. Since $ \sqrt{49}=7 $ and $ 7=3 + 4 $, we conclude $ \sqrt{49}=3 + 4 $ by the transitive property of equality.
# Answer: Transitive Property of Equality

### Problem 10: Identify the property for $ - 28=-28 $
# Explanation:
## Step1: Recall properties of equality
The reflexive property of equality states that for any real number $ a $, $ a=a $.
## Step2: Match with the given equation
Here, $ a=-28 $, so $ -28=-28 $ is an example of the reflexive property of equality.
# Answer: Reflexive Property of Equality

### Problem 11: Identify the property for $ 8x^{2}\cdot1 = 8x^{2} $
# Explanation:
## Step1: Recall properties of multiplication
The multiplicative identity property states that for any real number $ a $, $ a\times1=a $.
## Step2: Match with the given equation
Here, $ a = 8x^{2} $, and when we multiply $ 8x^{2} $ by 1, we get $ 8x^{2} $ back, so this is the multiplicative identity property.
# Answer: Multiplicative Identity Property

### Problem 12: Identify the property for $ 10y+(- 10y)=0 $
# Explanation:
## Step1: Recall properties of addition
The additive inverse property states that for any real number $ a $, $ a+(-a)=0 $.
## Step2: Match with the given equation
Here, $ a = 10y $, and $ -a=-10y $, so $ 10y+(-10y)=0 $ is an example of the additive inverse property.
# Answer: Additive Inverse Property

### Problem 13: Identify the property for $ (a + 4)\cdot0=0 $
# Explanation:
## Step1: Recall properties of multiplication
The zero - product property (or multiplicative property of zero) states that for any real number $ a $, $ a\times0 = 0 $.
## Step2: Match with the given equation
Here, $ a=(a + 4) $, so $ (a + 4)\cdot0=0 $ is an example of the multiplicative property of zero (zero - product property).
# Answer: Multiplicative Property of Zero (Zero - Product Property)

### Problem 14: Identify the property for $ -5(x + 7)=-5x-35 $
# Explanation:
## Step1: Recall properties of multiplication
The distributive property of multiplication over addition states that for any real numbers $ a $, $ b $, and $ c $, $ a(b + c)=ab+ac $.
## Step2: Match with the given equation
Here, $ a=-5 $, $ b=x $, and $ c = 7 $. So $ -5(x + 7)=-5\times x+(-5)\times7=-5x-35 $, which is the distributive property.
# Answer: Distributive Property of Multiplication over Addition

### Problem 15: Identify the property for $ (x + 2)+y=(2 + x)+y $
# Explanation:
## Step1: Recall properties of addition
The commutative property of addition states that for any real numbers $ a $ and $ b $, $ a + b=b + a $.
## Step2: Match with the given equation
In the equation $ (x + 2)+y=(2 + x)+y $, we are only re - arranging the order of $ x $ and $ 2 $ within the first parentheses, which is an application of the commutative property of addition (since $ x+2 = 2 + x $).
# Answer: Commutative Property of Addition

### Problem 16: Identify the property for "If $ x=-1 $, then $ -1=x $"
# Explanation:
## Step1: Recall properties of equality
The symmetric property of equality states that if $ a = b $, then $ b=a $.
## Step2: Match with the given statement
Here, $ a=x $ and $ b=-1 $. Since $ x=-1 $, we can say $ -1=x $ by the symmetric property of equality.
# Answer: Symmetric Property of Equality

### Real Number System Placement:
#### Irrational Numbers:
A number is irrational if it cannot be written as a fraction $ \frac{p}{q} $ where $ p $ and $ q $ are integers and $ q
eq0 $, and it has a non - repeating, non - terminating decimal expansion.
- $ D. \pi\approx3.14159\cdots $ (non - repeating, non - terminating decimal)
- $ I.-\sqrt{72}=-6\sqrt{2}\approx - 8.485\cdots $ ( $ \sqrt{2} $ is irrational, so $ - 6\sqrt{2} $ is irrational)

#### Rational Numbers (excluding Integers, Whole, Natural for now):
A rational number can be written as $ \frac{p}{q} $, $ p,q\in\mathbb{Z},q
eq0 $.
- $ A.-0.\overline{2}=-\frac{2}{9} $ (repeating decimal, rational)
- $ F.2\frac{1}{6}=\frac{13}{6} $ (fraction, rational)
- $ H.4.03=\frac{403}{100} $ (terminating decimal, rational)
- $ J.\sqrt{\frac{4}{9}}=\frac{2}{3} $ (square root of a fraction, rational)
- $ K.\frac{36}{9} = 4 $ (we will classify it further in integers)

#### Integers (excluding Whole, Natural for now):
Integers are $ \cdots,-3,-2,-1,0,1,2,3,\cdots $
- $ C.-\sqrt{100}=-10 $ (integer)
- $ G.-5 $ (integer)

#### Whole Numbers (excluding Natural for now):
Whole numbers are $ 0,1,2,3,\cdots $
- $ E.0 $ (whole number)

#### Natural Numbers:
Natural numbers are $ 1,2,3,\cdots $
- $ B.18 $ (natural number)
- $ K.\frac{36}{9}=4 $ (natural number, since $ 4\in\mathbb{N} $)

So:
- Irrational: D, I
- Rational (non - integer): A, F, H, J
- Integers (non - whole): C, G
- Whole (non - natural): E
- Natural: B, K

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 7: Identify the property for \( 4 + (x + y) = (4 + x) + y \)

Step1: Recall properties of addition

Step2: Match with the given equation

Step1: Recall properties of multiplication

Step2: Confirm the property

Step1: Recall properties of equality

Step2: Match with the given statement

Answer:

Problem 8: Identify the property for \( \frac{2}{5}\cdot\frac{5}{2}=1 \)