Math worksheet focusing on inverse properties and real number properties.

Problem Analysis:

The task involves three main sections:
1. Finding the additive inverse of each number.
2. Finding the multiplicative inverse of each number.
3. Identifying which property of real numbers is applied in each statement.

Let's solve each section step by step.

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Section 1: Additive Inverse

The additive inverse of a number \( x \) is the number that, when added to \( x \), results in zero. Mathematically, the additive inverse of \( x \) is \( -x \).

#### Solutions:
1. \( 15 \): The additive inverse is \( -15 \).
2. \( -27 \): The additive inverse is \( 27 \).
3. \( \frac{7}{9} \): The additive inverse is \( -\frac{7}{9} \).
4. \( -\frac{9}{16} \): The additive inverse is \( \frac{9}{16} \).
5. \( 0 \): The additive inverse is \( 0 \) (since \( 0 + 0 = 0 \)).
6. \( -1 \): The additive inverse is \( 1 \).

#### Final Answers for Section 1:
\[
\boxed{-15, 27, -\frac{7}{9}, \frac{9}{16}, 0, 1}
\]

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Section 2: Multiplicative Inverse

The multiplicative inverse of a number \( x \) is the number that, when multiplied by \( x \), results in 1. Mathematically, the multiplicative inverse of \( x \) is \( \frac{1}{x} \) (for \( x \neq 0 \)).

#### Solutions:
7. \( 15 \): The multiplicative inverse is \( \frac{1}{15} \).
8. \( -27 \): The multiplicative inverse is \( -\frac{1}{27} \).
9. \( \frac{7}{9} \): The multiplicative inverse is \( \frac{9}{7} \).
10. \( -\frac{9}{16} \): The multiplicative inverse is \( -\frac{16}{9} \).
11. \( 1 \): The multiplicative inverse is \( 1 \) (since \( 1 \times 1 = 1 \)).
12. \( \frac{1}{8} \): The multiplicative inverse is \( 8 \).

#### Final Answers for Section 2:
\[
\boxed{\frac{1}{15}, -\frac{1}{27}, \frac{9}{7}, -\frac{16}{9}, 1, 8}
\]

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Section 3: Identifying Properties of Real Numbers

We need to identify which property of real numbers is applied in each given statement. The properties are:
- Closure property of Addition or Multiplication
- Commutative property of Addition or Multiplication
- Associative property of Addition or Multiplication
- Identity property of Addition or Multiplication
- Inverse property of Addition or Multiplication
- Distributive property
- Zero Property

#### Solutions:
13. \( 43 + 25 = 25 + 43 \):
- This shows that the order of addition does not matter. It uses the Commutative property of Addition.
- Answer: Commutative property of Addition.

14. \( (8 \cdot 5) \cdot 10 = 8 \cdot (5 \cdot 10) \):
- This shows that the grouping of multiplication does not matter. It uses the Associative property of Multiplication.
- Answer: Associative property of Multiplication.

15. \( 3 + 5 + 8 = 3 + 8 + 5 \):
- This shows that the order of addition does not matter. It uses the Commutative property of Addition.
- Answer: Commutative property of Addition.

16. \( 5 + (6 + 2) = (5 + 6) + 2 \):
- This shows that the grouping of addition does not matter. It uses the Associative property of Addition.
- Answer: Associative property of Addition.

17. \( 12 \cdot 5 = 5 \cdot 12 \):
- This shows that the order of multiplication does not matter. It uses the Commutative property of Multiplication.
- Answer: Commutative property of Multiplication.

18. \( 4(5 + 3) = 4 \cdot 5 + 4 \cdot 3 \):
- This shows the distribution of multiplication over addition. It uses the Distributive property.
- Answer: Distributive property.

19. \( 3 \cdot 5 - 3 \cdot 2 = 3(5 - 2) \):
- This shows the reverse application of the distributive property. It uses the Distributive property.
- Answer: Distributive property.

20. \( 7 + (5 + 3) = 7 + (3 + 5) \):
- This shows that the grouping of addition does not matter. It uses the Associative property of Addition.
- Answer: Associative property of Addition.

21. \( -47 + 47 = 47 + (-47) = 0 \):
- This shows the use of the additive inverse to result in zero. It uses the Inverse property of Addition.
- Answer: Inverse property of Addition.

22. \( 2 \cdot 5 \cdot 4 = 2 \cdot 7 \cdot 4 \):
- This shows that the grouping of multiplication does not matter. It uses the Associative property of Multiplication.
- Answer: Associative property of Multiplication.

23. \( 7 + (-7) = 0 \):
- This shows the use of the additive inverse to result in zero. It uses the Inverse property of Addition.
- Answer: Inverse property of Addition.

#### Final Answers for Section 3:
\[
\boxed{
\text{Commutative property of Addition, Associative property of Multiplication, Commutative property of Addition, Associative property of Addition, Commutative property of Multiplication, Distributive property, Distributive property, Associative property of Addition, Inverse property of Addition, Associative property of Multiplication, Inverse property of Addition}
}
\]

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Final Summary of All Answers:

1. Additive inverses: \(\boxed{-15, 27, -\frac{7}{9}, \frac{9}{16}, 0, 1}\)
2. Multiplicative inverses: \(\boxed{\frac{1}{15}, -\frac{1}{27}, \frac{9}{7}, -\frac{16}{9}, 1, 8}\)
3. Properties of real numbers: \(\boxed{
\text{Commutative property of Addition, Associative property of Multiplication, Commutative property of Addition, Associative property of Addition, Commutative property of Multiplication, Distributive property, Distributive property, Associative property of Addition, Inverse property of Addition, Associative property of Multiplication, Inverse property of Addition}
}\)