Sure, let's look at each of the given equations and identify the property they illustrate step-by-step.
8. [tex]\(-\frac{1}{3} + \frac{1}{3} = 0\)[/tex]
- This equation demonstrates that adding a number to its opposite (additive inverse) equals zero.
- Property: Additive Inverse Property
9. [tex]\((9 \cdot -4) \cdot 7 = 7 \cdot (9 \cdot -4)\)[/tex]
- This equation shows that the grouping (association) of numbers in multiplication does not change the product.
- Property: Associative Property of Multiplication
10. [tex]\(6(2x - 1) = 6 \cdot 2x - 6 \cdot 1\)[/tex]
- This equation illustrates that a number can be distributed over a sum or difference inside parentheses.
- Property: Distributive Property
11. [tex]\(5x^2 \cdot 1 = 5x^2\)[/tex]
- This equation shows that any number multiplied by one remains unchanged.
- Property: Identity Property of Multiplication
12. [tex]\((8a + 2b) + c = 8a + (2b + c)\)[/tex]
- This equation illustrates that the grouping of numbers in addition does not change the sum.
- Property: Associative Property of Addition
13. [tex]\(\frac{8x}{3} \cdot \frac{3}{8x} = 1\)[/tex]
- This equation demonstrates that a number multiplied by its multiplicative inverse (reciprocal) equals one.
- Property: Multiplicative Inverse Property
14. Name the additive inverse of [tex]\(\sqrt{52}\)[/tex]
- The additive inverse of a number is the number that, when added to the original number, results in zero. So, the additive inverse of [tex]\(\sqrt{52}\)[/tex] is [tex]\(-\sqrt{52}\)[/tex].
- Value: [tex]\(-7.211102550927978\)[/tex]
These steps give a comprehensive walkthrough of each mathematical property illustrated by the given equations.
