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\begin{tabular}{|c|c|c|} \hline \multicolumn{3}{|c|}{Properties of Real Numbers} \\ \hline Property & Addition Example & Multiplication Example \\ \hline \multicolumn{3}{|l|}{Commutative} \\ \hline \multicolumn{3}{|l|}{Associative} \\ \hline \multicolumn{3}{|l|}{Identity} \\ \hline \multicolumn{3}{|l|}{Inverse} \\ \hline Distributive & & \\ \hline \end{tabular}
Let's identify each property of real numbers based on the given expressions.
1. Statement: [tex]\(5 \cdot(2 \cdot 8)=(5 \cdot 2) \cdot 8\)[/tex] - Property: Associative Property of Multiplication - This property states that the way in which numbers are grouped in multiplication does not change their product.
2. Statement: [tex]\(6(3-7)=6 \cdot 3-6 \cdot 7\)[/tex] - Property: Distributive Property - This property allows us to multiply a single term across a sum or difference inside parentheses.
3. Statement: [tex]\(7y+(-7y)=0\)[/tex] - Property: Inverse Property of Addition - This property states that every number has an additive inverse (opposite) that sums to zero.
4. Statement: [tex]\(3a + 2b = 2b + 3a\)[/tex] - Property: Commutative Property of Addition - This property states that the order of addition does not affect the sum.
5. Statement: [tex]\((x^2 + 8x) + 1 = x^2 + (8x + 1)\)[/tex] - Property: Associative Property of Addition - This property states that the way in which numbers are grouped in addition does not change their sum.
6. Statement: [tex]\(\frac{1}{5} \cdot 5 = 1\)[/tex] - Property: Identity Property of Multiplication - This property states that any number multiplied by 1 remains the same.
7. Statement: [tex]\(3m + 3n = 3(m + n)\)[/tex] - Property: Distributive Property - This property allows us to factor out a common factor from the terms in addition.
8. Statement: [tex]\(9c + 0 = 9c\)[/tex] - Property: Identity Property of Addition - This property states that any number added to zero remains the same.
Summarized, the properties corresponding to the problems are:
- 13: Associative Property of Multiplication - 14: Distributive Property - 15: Inverse Property of Addition - 16: Commutative Property of Addition - 17: Associative Property of Addition - 18: Identity Property of Multiplication - 19: Distributive Property - 20: Identity Property of Addition
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